TPTP Problem File: SLH0525^1.p
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%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : Youngs_Inequality/0000_Youngs/prob_00504_021313__13110724_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1432 ( 525 unt; 156 typ; 0 def)
% Number of atoms : 3975 (1429 equ; 0 cnn)
% Maximal formula atoms : 11 ( 3 avg)
% Number of connectives : 11809 ( 325 ~; 77 |; 274 &;9429 @)
% ( 0 <=>;1704 =>; 0 <=; 0 <~>)
% Maximal formula depth : 21 ( 7 avg)
% Number of types : 13 ( 12 usr)
% Number of type conns : 831 ( 831 >; 0 *; 0 +; 0 <<)
% Number of symbols : 147 ( 144 usr; 20 con; 0-4 aty)
% Number of variables : 3663 ( 163 ^;3396 !; 104 ?;3663 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 16:31:08.284
%------------------------------------------------------------------------------
% Could-be-implicit typings (12)
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Real__Oreal_J_J_J,type,
set_set_set_real: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Real__Oreal_J_J,type,
set_set_real: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
set_set_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Int__Oint_J_J,type,
set_set_int: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_I_Eo_J_J,type,
set_set_o: $tType ).
thf(ty_n_t__Set__Oset_It__Real__Oreal_J,type,
set_real: $tType ).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
set_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Int__Oint_J,type,
set_int: $tType ).
thf(ty_n_t__Set__Oset_I_Eo_J,type,
set_o: $tType ).
thf(ty_n_t__Real__Oreal,type,
real: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_t__Int__Oint,type,
int: $tType ).
% Explicit typings (144)
thf(sy_c_Complete__Lattices_OInf__class_OInf_001_Eo,type,
complete_Inf_Inf_o: set_o > $o ).
thf(sy_c_Complete__Lattices_OInf__class_OInf_001t__Int__Oint,type,
complete_Inf_Inf_int: set_int > int ).
thf(sy_c_Complete__Lattices_OInf__class_OInf_001t__Nat__Onat,type,
complete_Inf_Inf_nat: set_nat > nat ).
thf(sy_c_Complete__Lattices_OInf__class_OInf_001t__Real__Oreal,type,
comple4887499456419720421f_real: set_real > real ).
thf(sy_c_Complete__Lattices_OInf__class_OInf_001t__Set__Oset_I_Eo_J,type,
comple3063163877087187839_set_o: set_set_o > set_o ).
thf(sy_c_Complete__Lattices_OInf__class_OInf_001t__Set__Oset_It__Nat__Onat_J,type,
comple7806235888213564991et_nat: set_set_nat > set_nat ).
thf(sy_c_Complete__Lattices_OInf__class_OInf_001t__Set__Oset_It__Real__Oreal_J,type,
comple8289635161444856091t_real: set_set_real > set_real ).
thf(sy_c_Complete__Lattices_OInf__class_OInf_001t__Set__Oset_It__Set__Oset_It__Real__Oreal_J_J,type,
comple6920828426275262033t_real: set_set_set_real > set_set_real ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001_Eo,type,
complete_Sup_Sup_o: set_o > $o ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Int__Oint,type,
complete_Sup_Sup_int: set_int > int ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Nat__Onat,type,
complete_Sup_Sup_nat: set_nat > nat ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Real__Oreal,type,
comple1385675409528146559p_real: set_real > real ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_I_Eo_J,type,
comple90263536869209701_set_o: set_set_o > set_o ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__Int__Oint_J,type,
comple3221217463730067765et_int: set_set_int > set_int ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__Nat__Onat_J,type,
comple7399068483239264473et_nat: set_set_nat > set_nat ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__Real__Oreal_J,type,
comple3096694443085538997t_real: set_set_real > set_real ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__Set__Oset_It__Real__Oreal_J_J,type,
comple5917660045593844715t_real: set_set_set_real > set_set_real ).
thf(sy_c_Fun_Ocomp_001t__Real__Oreal_001t__Real__Oreal_001t__Real__Oreal,type,
comp_real_real_real: ( real > real ) > ( real > real ) > real > real ).
thf(sy_c_Fun_Omonotone__on_001t__Real__Oreal_001t__Real__Oreal,type,
monoto4017252874604999745l_real: set_real > ( real > real > $o ) > ( real > real > $o ) > ( real > real ) > $o ).
thf(sy_c_Groups_Oabs__class_Oabs_001t__Int__Oint,type,
abs_abs_int: int > int ).
thf(sy_c_Groups_Oabs__class_Oabs_001t__Real__Oreal,type,
abs_abs_real: real > real ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Int__Oint,type,
minus_minus_int: int > int > int ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
minus_minus_nat: nat > nat > nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Real__Oreal,type,
minus_minus_real: real > real > real ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Int__Oint_J,type,
minus_minus_set_int: set_int > set_int > set_int ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Nat__Onat_J,type,
minus_minus_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Real__Oreal_J,type,
minus_minus_set_real: set_real > set_real > set_real ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Set__Oset_It__Real__Oreal_J_J,type,
minus_5467046032205032049t_real: set_set_real > set_set_real > set_set_real ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Int__Oint,type,
times_times_int: int > int > int ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Nat__Onat,type,
times_times_nat: nat > nat > nat ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Real__Oreal,type,
times_times_real: real > real > real ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Int__Oint,type,
zero_zero_int: int ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
zero_zero_nat: nat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Real__Oreal,type,
zero_zero_real: real ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001_Eo_001t__Int__Oint,type,
groups8505340233167759370_o_int: ( $o > int ) > set_o > int ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001_Eo_001t__Nat__Onat,type,
groups8507830703676809646_o_nat: ( $o > nat ) > set_o > nat ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001_Eo_001t__Real__Oreal,type,
groups8691415230153176458o_real: ( $o > real ) > set_o > real ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Nat__Onat_001t__Int__Oint,type,
groups3539618377306564664at_int: ( nat > int ) > set_nat > int ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Nat__Onat_001t__Nat__Onat,type,
groups3542108847815614940at_nat: ( nat > nat ) > set_nat > nat ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Nat__Onat_001t__Real__Oreal,type,
groups6591440286371151544t_real: ( nat > real ) > set_nat > real ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Real__Oreal_001t__Int__Oint,type,
groups1932886352136224148al_int: ( real > int ) > set_real > int ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Real__Oreal_001t__Nat__Onat,type,
groups1935376822645274424al_nat: ( real > nat ) > set_real > nat ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Real__Oreal_001t__Real__Oreal,type,
groups8097168146408367636l_real: ( real > real ) > set_real > real ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Set__Oset_It__Real__Oreal_J_001t__Int__Oint,type,
groups3009712052913938890al_int: ( set_real > int ) > set_set_real > int ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Set__Oset_It__Real__Oreal_J_001t__Nat__Onat,type,
groups3012202523422989166al_nat: ( set_real > nat ) > set_set_real > nat ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Set__Oset_It__Real__Oreal_J_001t__Real__Oreal,type,
groups8702937949983641418l_real: ( set_real > real ) > set_set_real > real ).
thf(sy_c_Henstock__Kurzweil__Integration_Ohas__integral_001t__Real__Oreal_001t__Real__Oreal,type,
hensto240673015341029504l_real: ( real > real ) > real > set_real > $o ).
thf(sy_c_Henstock__Kurzweil__Integration_Ointegrable__on_001t__Real__Oreal_001t__Real__Oreal,type,
hensto5963834015518849588l_real: ( real > real ) > set_real > $o ).
thf(sy_c_If_001t__Nat__Onat,type,
if_nat: $o > nat > nat > nat ).
thf(sy_c_If_001t__Real__Oreal,type,
if_real: $o > real > real > real ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Int__Oint,type,
semiri1314217659103216013at_int: nat > int ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Nat__Onat,type,
semiri1316708129612266289at_nat: nat > nat ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Real__Oreal,type,
semiri5074537144036343181t_real: nat > real ).
thf(sy_c_Orderings_Obot__class_Obot_001_Eo,type,
bot_bot_o: $o ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Nat__Onat,type,
bot_bot_nat: nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_I_Eo_J,type,
bot_bot_set_o: set_o ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Int__Oint_J,type,
bot_bot_set_int: set_int ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Nat__Onat_J,type,
bot_bot_set_nat: set_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Real__Oreal_J,type,
bot_bot_set_real: set_real ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
bot_bot_set_set_nat: set_set_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__Real__Oreal_J_J,type,
bot_bot_set_set_real: set_set_real ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Real__Oreal_J_J_J,type,
bot_bo3378928929837779682t_real: set_set_set_real ).
thf(sy_c_Orderings_Oord__class_Oless_001_Eo,type,
ord_less_o: $o > $o > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Int__Oint,type,
ord_less_int: int > int > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
ord_less_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Real__Oreal,type,
ord_less_real: real > real > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Int__Oint_J,type,
ord_less_set_int: set_int > set_int > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Real__Oreal_J,type,
ord_less_set_real: set_real > set_real > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Set__Oset_It__Real__Oreal_J_J,type,
ord_le7926960851185191020t_real: set_set_real > set_set_real > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_Eo,type,
ord_less_eq_o: $o > $o > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Int__Oint,type,
ord_less_eq_int: int > int > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Real__Oreal,type,
ord_less_eq_real: real > real > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_Eo_J,type,
ord_less_eq_set_o: set_o > set_o > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Int__Oint_J,type,
ord_less_eq_set_int: set_int > set_int > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_eq_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Real__Oreal_J,type,
ord_less_eq_set_real: set_real > set_real > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Real__Oreal_J_J,type,
ord_le3558479182127378552t_real: set_set_real > set_set_real > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Real__Oreal_J,type,
top_top_set_real: set_real ).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Int__Oint,type,
divide_divide_int: int > int > int ).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Nat__Onat,type,
divide_divide_nat: nat > nat > nat ).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Real__Oreal,type,
divide_divide_real: real > real > real ).
thf(sy_c_Set_OCollect_001_Eo,type,
collect_o: ( $o > $o ) > set_o ).
thf(sy_c_Set_OCollect_001t__Int__Oint,type,
collect_int: ( int > $o ) > set_int ).
thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(sy_c_Set_OCollect_001t__Real__Oreal,type,
collect_real: ( real > $o ) > set_real ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__Real__Oreal_J,type,
collect_set_real: ( set_real > $o ) > set_set_real ).
thf(sy_c_Set_Oimage_001_Eo_001_Eo,type,
image_o_o: ( $o > $o ) > set_o > set_o ).
thf(sy_c_Set_Oimage_001_Eo_001t__Nat__Onat,type,
image_o_nat: ( $o > nat ) > set_o > set_nat ).
thf(sy_c_Set_Oimage_001_Eo_001t__Real__Oreal,type,
image_o_real: ( $o > real ) > set_o > set_real ).
thf(sy_c_Set_Oimage_001_Eo_001t__Set__Oset_It__Real__Oreal_J,type,
image_o_set_real: ( $o > set_real ) > set_o > set_set_real ).
thf(sy_c_Set_Oimage_001t__Int__Oint_001t__Int__Oint,type,
image_int_int: ( int > int ) > set_int > set_int ).
thf(sy_c_Set_Oimage_001t__Int__Oint_001t__Set__Oset_It__Real__Oreal_J,type,
image_int_set_real: ( int > set_real ) > set_int > set_set_real ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001_Eo,type,
image_nat_o: ( nat > $o ) > set_nat > set_o ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Int__Oint,type,
image_nat_int: ( nat > int ) > set_nat > set_int ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Nat__Onat,type,
image_nat_nat: ( nat > nat ) > set_nat > set_nat ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Real__Oreal,type,
image_nat_real: ( nat > real ) > set_nat > set_real ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Set__Oset_It__Nat__Onat_J,type,
image_nat_set_nat: ( nat > set_nat ) > set_nat > set_set_nat ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Set__Oset_It__Real__Oreal_J,type,
image_nat_set_real: ( nat > set_real ) > set_nat > set_set_real ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Set__Oset_It__Set__Oset_It__Real__Oreal_J_J,type,
image_396256051147326063t_real: ( nat > set_set_real ) > set_nat > set_set_set_real ).
thf(sy_c_Set_Oimage_001t__Real__Oreal_001_Eo,type,
image_real_o: ( real > $o ) > set_real > set_o ).
thf(sy_c_Set_Oimage_001t__Real__Oreal_001t__Nat__Onat,type,
image_real_nat: ( real > nat ) > set_real > set_nat ).
thf(sy_c_Set_Oimage_001t__Real__Oreal_001t__Real__Oreal,type,
image_real_real: ( real > real ) > set_real > set_real ).
thf(sy_c_Set_Oimage_001t__Real__Oreal_001t__Set__Oset_It__Nat__Onat_J,type,
image_real_set_nat: ( real > set_nat ) > set_real > set_set_nat ).
thf(sy_c_Set_Oimage_001t__Real__Oreal_001t__Set__Oset_It__Real__Oreal_J,type,
image_real_set_real: ( real > set_real ) > set_real > set_set_real ).
thf(sy_c_Set_Oimage_001t__Real__Oreal_001t__Set__Oset_It__Set__Oset_It__Real__Oreal_J_J,type,
image_3243600997494576203t_real: ( real > set_set_real ) > set_real > set_set_set_real ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Int__Oint_J,type,
image_3739036796817536367et_int: ( set_nat > set_int ) > set_set_nat > set_set_int ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Real__Oreal_J_001_Eo,type,
image_set_real_o: ( set_real > $o ) > set_set_real > set_o ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Real__Oreal_J_001t__Nat__Onat,type,
image_set_real_nat: ( set_real > nat ) > set_set_real > set_nat ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Real__Oreal_J_001t__Real__Oreal,type,
image_set_real_real: ( set_real > real ) > set_set_real > set_real ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Real__Oreal_J_001t__Set__Oset_It__Nat__Onat_J,type,
image_7270232309134952815et_nat: ( set_real > set_nat ) > set_set_real > set_set_nat ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Real__Oreal_J_001t__Set__Oset_It__Real__Oreal_J,type,
image_2436557299294012491t_real: ( set_real > set_real ) > set_set_real > set_set_real ).
thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001_Eo,type,
set_or8904488021354931149Most_o: $o > $o > set_o ).
thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Int__Oint,type,
set_or1266510415728281911st_int: int > int > set_int ).
thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Nat__Onat,type,
set_or1269000886237332187st_nat: nat > nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Real__Oreal,type,
set_or1222579329274155063t_real: real > real > set_real ).
thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Set__Oset_It__Real__Oreal_J,type,
set_or7743017856606604397t_real: set_real > set_real > set_set_real ).
thf(sy_c_Set__Interval_Oord__class_OatLeast_001t__Real__Oreal,type,
set_ord_atLeast_real: real > set_real ).
thf(sy_c_Tagged__Division_Odivision__of_001t__Real__Oreal,type,
tagged6100619406677346166f_real: set_set_real > set_real > $o ).
thf(sy_c_Topological__Spaces_Ocontinuous__on_001t__Real__Oreal_001t__Real__Oreal,type,
topolo5044208981011980120l_real: set_real > ( real > real ) > $o ).
thf(sy_c_Topological__Spaces_Ouniformly__continuous__on_001t__Real__Oreal_001t__Real__Oreal,type,
topolo8845477368217174713l_real: set_real > ( real > real ) > $o ).
thf(sy_c_Youngs_Oregular__division,type,
regular_division: real > real > nat > set_set_real ).
thf(sy_c_member_001_Eo,type,
member_o: $o > set_o > $o ).
thf(sy_c_member_001t__Int__Oint,type,
member_int: int > set_int > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Real__Oreal,type,
member_real: real > set_real > $o ).
thf(sy_c_member_001t__Set__Oset_I_Eo_J,type,
member_set_o: set_o > set_set_o > $o ).
thf(sy_c_member_001t__Set__Oset_It__Nat__Onat_J,type,
member_set_nat: set_nat > set_set_nat > $o ).
thf(sy_c_member_001t__Set__Oset_It__Real__Oreal_J,type,
member_set_real: set_real > set_set_real > $o ).
thf(sy_c_member_001t__Set__Oset_It__Set__Oset_It__Real__Oreal_J_J,type,
member_set_set_real: set_set_real > set_set_set_real > $o ).
thf(sy_v__092_060delta_062____,type,
delta: real ).
thf(sy_v__092_060epsilon_062____,type,
epsilon: real ).
thf(sy_v_a,type,
a: real ).
thf(sy_v_a__seg____,type,
a_seg: real > real ).
thf(sy_v_b,type,
b: real ).
thf(sy_v_del____,type,
del: real > real ).
thf(sy_v_f,type,
f: real > real ).
thf(sy_v_f1____,type,
f1: real > real ).
thf(sy_v_f2____,type,
f2: real > real ).
thf(sy_v_g,type,
g: real > real ).
thf(sy_v_lower____,type,
lower: real > real ).
thf(sy_v_n____,type,
n: nat ).
thf(sy_v_upper____,type,
upper: real > real ).
% Relevant facts (1270)
thf(fact_0_False,axiom,
a != zero_zero_real ).
% False
thf(fact_1_f_I1_J,axiom,
( ( f @ zero_zero_real )
= zero_zero_real ) ).
% f(1)
thf(fact_2_a,axiom,
ord_less_eq_real @ zero_zero_real @ a ).
% a
thf(fact_3_f_I2_J,axiom,
( ( f @ a )
= b ) ).
% f(2)
thf(fact_4_f__iff_I2_J,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ( ord_less_eq_real @ ( f @ X ) @ ( f @ Y ) )
= ( ord_less_eq_real @ X @ Y ) ) ) ) ).
% f_iff(2)
thf(fact_5_a__seg__def,axiom,
( a_seg
= ( ^ [U: real] : ( divide_divide_real @ ( times_times_real @ U @ a ) @ ( semiri5074537144036343181t_real @ n ) ) ) ) ).
% a_seg_def
thf(fact_6_g,axiom,
! [X: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ X @ a )
=> ( ( g @ ( f @ X ) )
= X ) ) ) ).
% g
thf(fact_7__092_060open_062_092_060delta_062_A_092_060le_062_Aa_092_060close_062,axiom,
ord_less_eq_real @ delta @ a ).
% \<open>\<delta> \<le> a\<close>
thf(fact_8_f1__lower,axiom,
! [X: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ X @ a )
=> ( ord_less_eq_real @ ( f1 @ X ) @ ( f @ X ) ) ) ) ).
% f1_lower
thf(fact_9_f2__upper,axiom,
! [X: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ X @ a )
=> ( ord_less_eq_real @ ( f @ X ) @ ( f2 @ X ) ) ) ) ).
% f2_upper
thf(fact_10__092_060open_0620_A_092_060le_062_Ab_092_060close_062,axiom,
ord_less_eq_real @ zero_zero_real @ b ).
% \<open>0 \<le> b\<close>
thf(fact_11_sum__abs__ge__zero,axiom,
! [F: set_real > real,A: set_set_real] :
( ord_less_eq_real @ zero_zero_real
@ ( groups8702937949983641418l_real
@ ^ [I: set_real] : ( abs_abs_real @ ( F @ I ) )
@ A ) ) ).
% sum_abs_ge_zero
thf(fact_12_divide__le__0__abs__iff,axiom,
! [A2: real,B: real] :
( ( ord_less_eq_real @ ( divide_divide_real @ A2 @ ( abs_abs_real @ B ) ) @ zero_zero_real )
= ( ( ord_less_eq_real @ A2 @ zero_zero_real )
| ( B = zero_zero_real ) ) ) ).
% divide_le_0_abs_iff
thf(fact_13_zero__le__divide__abs__iff,axiom,
! [A2: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ A2 @ ( abs_abs_real @ B ) ) )
= ( ( ord_less_eq_real @ zero_zero_real @ A2 )
| ( B = zero_zero_real ) ) ) ).
% zero_le_divide_abs_iff
thf(fact_14_sum__abs,axiom,
! [F: set_real > real,A: set_set_real] :
( ord_less_eq_real @ ( abs_abs_real @ ( groups8702937949983641418l_real @ F @ A ) )
@ ( groups8702937949983641418l_real
@ ^ [I: set_real] : ( abs_abs_real @ ( F @ I ) )
@ A ) ) ).
% sum_abs
thf(fact_15_abs__of__nonneg,axiom,
! [A2: real] :
( ( ord_less_eq_real @ zero_zero_real @ A2 )
=> ( ( abs_abs_real @ A2 )
= A2 ) ) ).
% abs_of_nonneg
thf(fact_16_abs__of__nonneg,axiom,
! [A2: int] :
( ( ord_less_eq_int @ zero_zero_int @ A2 )
=> ( ( abs_abs_int @ A2 )
= A2 ) ) ).
% abs_of_nonneg
thf(fact_17_abs__le__self__iff,axiom,
! [A2: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ A2 ) @ A2 )
= ( ord_less_eq_real @ zero_zero_real @ A2 ) ) ).
% abs_le_self_iff
thf(fact_18_abs__le__self__iff,axiom,
! [A2: int] :
( ( ord_less_eq_int @ ( abs_abs_int @ A2 ) @ A2 )
= ( ord_less_eq_int @ zero_zero_int @ A2 ) ) ).
% abs_le_self_iff
thf(fact_19_abs__le__zero__iff,axiom,
! [A2: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ A2 ) @ zero_zero_real )
= ( A2 = zero_zero_real ) ) ).
% abs_le_zero_iff
thf(fact_20_abs__le__zero__iff,axiom,
! [A2: int] :
( ( ord_less_eq_int @ ( abs_abs_int @ A2 ) @ zero_zero_int )
= ( A2 = zero_zero_int ) ) ).
% abs_le_zero_iff
thf(fact_21_mult__divide__mult__cancel__left__if,axiom,
! [C: real,A2: real,B: real] :
( ( ( C = zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ C @ A2 ) @ ( times_times_real @ C @ B ) )
= zero_zero_real ) )
& ( ( C != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ C @ A2 ) @ ( times_times_real @ C @ B ) )
= ( divide_divide_real @ A2 @ B ) ) ) ) ).
% mult_divide_mult_cancel_left_if
thf(fact_22_nonzero__mult__divide__mult__cancel__left,axiom,
! [C: real,A2: real,B: real] :
( ( C != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ C @ A2 ) @ ( times_times_real @ C @ B ) )
= ( divide_divide_real @ A2 @ B ) ) ) ).
% nonzero_mult_divide_mult_cancel_left
thf(fact_23_abs__idempotent,axiom,
! [A2: real] :
( ( abs_abs_real @ ( abs_abs_real @ A2 ) )
= ( abs_abs_real @ A2 ) ) ).
% abs_idempotent
thf(fact_24_a__seg__eq__a__iff,axiom,
! [X: real] :
( ( ( a_seg @ X )
= a )
= ( X
= ( semiri5074537144036343181t_real @ n ) ) ) ).
% a_seg_eq_a_iff
thf(fact_25_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_26_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A2: real] :
( ( minus_minus_real @ A2 @ A2 )
= zero_zero_real ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_27_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A2: nat] :
( ( minus_minus_nat @ A2 @ A2 )
= zero_zero_nat ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_28_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A2: int] :
( ( minus_minus_int @ A2 @ A2 )
= zero_zero_int ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_29_diff__zero,axiom,
! [A2: real] :
( ( minus_minus_real @ A2 @ zero_zero_real )
= A2 ) ).
% diff_zero
thf(fact_30_diff__zero,axiom,
! [A2: nat] :
( ( minus_minus_nat @ A2 @ zero_zero_nat )
= A2 ) ).
% diff_zero
thf(fact_31_diff__zero,axiom,
! [A2: int] :
( ( minus_minus_int @ A2 @ zero_zero_int )
= A2 ) ).
% diff_zero
thf(fact_32_zero__diff,axiom,
! [A2: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A2 )
= zero_zero_nat ) ).
% zero_diff
thf(fact_33_diff__0__right,axiom,
! [A2: real] :
( ( minus_minus_real @ A2 @ zero_zero_real )
= A2 ) ).
% diff_0_right
thf(fact_34_diff__0__right,axiom,
! [A2: int] :
( ( minus_minus_int @ A2 @ zero_zero_int )
= A2 ) ).
% diff_0_right
thf(fact_35_diff__self,axiom,
! [A2: real] :
( ( minus_minus_real @ A2 @ A2 )
= zero_zero_real ) ).
% diff_self
thf(fact_36_diff__self,axiom,
! [A2: int] :
( ( minus_minus_int @ A2 @ A2 )
= zero_zero_int ) ).
% diff_self
thf(fact_37_division__ring__divide__zero,axiom,
! [A2: real] :
( ( divide_divide_real @ A2 @ zero_zero_real )
= zero_zero_real ) ).
% division_ring_divide_zero
thf(fact_38_divide__cancel__right,axiom,
! [A2: real,C: real,B: real] :
( ( ( divide_divide_real @ A2 @ C )
= ( divide_divide_real @ B @ C ) )
= ( ( C = zero_zero_real )
| ( A2 = B ) ) ) ).
% divide_cancel_right
thf(fact_39_divide__cancel__left,axiom,
! [C: real,A2: real,B: real] :
( ( ( divide_divide_real @ C @ A2 )
= ( divide_divide_real @ C @ B ) )
= ( ( C = zero_zero_real )
| ( A2 = B ) ) ) ).
% divide_cancel_left
thf(fact_40_divide__eq__0__iff,axiom,
! [A2: real,B: real] :
( ( ( divide_divide_real @ A2 @ B )
= zero_zero_real )
= ( ( A2 = zero_zero_real )
| ( B = zero_zero_real ) ) ) ).
% divide_eq_0_iff
thf(fact_41_times__divide__eq__right,axiom,
! [A2: real,B: real,C: real] :
( ( times_times_real @ A2 @ ( divide_divide_real @ B @ C ) )
= ( divide_divide_real @ ( times_times_real @ A2 @ B ) @ C ) ) ).
% times_divide_eq_right
thf(fact_42_divide__divide__eq__right,axiom,
! [A2: real,B: real,C: real] :
( ( divide_divide_real @ A2 @ ( divide_divide_real @ B @ C ) )
= ( divide_divide_real @ ( times_times_real @ A2 @ C ) @ B ) ) ).
% divide_divide_eq_right
thf(fact_43_divide__divide__eq__left,axiom,
! [A2: real,B: real,C: real] :
( ( divide_divide_real @ ( divide_divide_real @ A2 @ B ) @ C )
= ( divide_divide_real @ A2 @ ( times_times_real @ B @ C ) ) ) ).
% divide_divide_eq_left
thf(fact_44_times__divide__eq__left,axiom,
! [B: real,C: real,A2: real] :
( ( times_times_real @ ( divide_divide_real @ B @ C ) @ A2 )
= ( divide_divide_real @ ( times_times_real @ B @ A2 ) @ C ) ) ).
% times_divide_eq_left
thf(fact_45_abs__zero,axiom,
( ( abs_abs_real @ zero_zero_real )
= zero_zero_real ) ).
% abs_zero
thf(fact_46_abs__zero,axiom,
( ( abs_abs_int @ zero_zero_int )
= zero_zero_int ) ).
% abs_zero
thf(fact_47_abs__eq__0,axiom,
! [A2: real] :
( ( ( abs_abs_real @ A2 )
= zero_zero_real )
= ( A2 = zero_zero_real ) ) ).
% abs_eq_0
thf(fact_48_abs__eq__0,axiom,
! [A2: int] :
( ( ( abs_abs_int @ A2 )
= zero_zero_int )
= ( A2 = zero_zero_int ) ) ).
% abs_eq_0
thf(fact_49_abs__0__eq,axiom,
! [A2: real] :
( ( zero_zero_real
= ( abs_abs_real @ A2 ) )
= ( A2 = zero_zero_real ) ) ).
% abs_0_eq
thf(fact_50_abs__0__eq,axiom,
! [A2: int] :
( ( zero_zero_int
= ( abs_abs_int @ A2 ) )
= ( A2 = zero_zero_int ) ) ).
% abs_0_eq
thf(fact_51_abs__divide,axiom,
! [A2: real,B: real] :
( ( abs_abs_real @ ( divide_divide_real @ A2 @ B ) )
= ( divide_divide_real @ ( abs_abs_real @ A2 ) @ ( abs_abs_real @ B ) ) ) ).
% abs_divide
thf(fact_52_fa__eq__b,axiom,
( ( f @ ( a_seg @ ( semiri5074537144036343181t_real @ n ) ) )
= b ) ).
% fa_eq_b
thf(fact_53_sum_Oneutral__const,axiom,
! [A: set_set_real] :
( ( groups8702937949983641418l_real
@ ^ [Uu: set_real] : zero_zero_real
@ A )
= zero_zero_real ) ).
% sum.neutral_const
thf(fact_54_abs__sum__abs,axiom,
! [F: set_real > real,A: set_set_real] :
( ( abs_abs_real
@ ( groups8702937949983641418l_real
@ ^ [A3: set_real] : ( abs_abs_real @ ( F @ A3 ) )
@ A ) )
= ( groups8702937949983641418l_real
@ ^ [A3: set_real] : ( abs_abs_real @ ( F @ A3 ) )
@ A ) ) ).
% abs_sum_abs
thf(fact_55_diff__ge__0__iff__ge,axiom,
! [A2: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ A2 @ B ) )
= ( ord_less_eq_real @ B @ A2 ) ) ).
% diff_ge_0_iff_ge
thf(fact_56_diff__ge__0__iff__ge,axiom,
! [A2: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( minus_minus_int @ A2 @ B ) )
= ( ord_less_eq_int @ B @ A2 ) ) ).
% diff_ge_0_iff_ge
thf(fact_57_mem__Collect__eq,axiom,
! [A2: set_real,P: set_real > $o] :
( ( member_set_real @ A2 @ ( collect_set_real @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_58_mem__Collect__eq,axiom,
! [A2: real,P: real > $o] :
( ( member_real @ A2 @ ( collect_real @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_59_mem__Collect__eq,axiom,
! [A2: nat,P: nat > $o] :
( ( member_nat @ A2 @ ( collect_nat @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_60_mem__Collect__eq,axiom,
! [A2: $o,P: $o > $o] :
( ( member_o @ A2 @ ( collect_o @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_61_Collect__mem__eq,axiom,
! [A: set_set_real] :
( ( collect_set_real
@ ^ [X2: set_real] : ( member_set_real @ X2 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_62_Collect__mem__eq,axiom,
! [A: set_real] :
( ( collect_real
@ ^ [X2: real] : ( member_real @ X2 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_63_Collect__mem__eq,axiom,
! [A: set_nat] :
( ( collect_nat
@ ^ [X2: nat] : ( member_nat @ X2 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_64_Collect__mem__eq,axiom,
! [A: set_o] :
( ( collect_o
@ ^ [X2: $o] : ( member_o @ X2 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_65_nonzero__mult__divide__mult__cancel__right2,axiom,
! [C: real,A2: real,B: real] :
( ( C != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ A2 @ C ) @ ( times_times_real @ C @ B ) )
= ( divide_divide_real @ A2 @ B ) ) ) ).
% nonzero_mult_divide_mult_cancel_right2
thf(fact_66_nonzero__mult__divide__mult__cancel__right,axiom,
! [C: real,A2: real,B: real] :
( ( C != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ A2 @ C ) @ ( times_times_real @ B @ C ) )
= ( divide_divide_real @ A2 @ B ) ) ) ).
% nonzero_mult_divide_mult_cancel_right
thf(fact_67_nonzero__mult__divide__mult__cancel__left2,axiom,
! [C: real,A2: real,B: real] :
( ( C != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ C @ A2 ) @ ( times_times_real @ B @ C ) )
= ( divide_divide_real @ A2 @ B ) ) ) ).
% nonzero_mult_divide_mult_cancel_left2
thf(fact_68_a__seg__le__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ ( a_seg @ X ) @ ( a_seg @ Y ) )
= ( ord_less_eq_real @ X @ Y ) ) ).
% a_seg_le_iff
thf(fact_69_a__seg__ge__0,axiom,
! [X: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( a_seg @ X ) )
= ( ord_less_eq_real @ zero_zero_real @ X ) ) ).
% a_seg_ge_0
thf(fact_70_a__seg__le__a,axiom,
! [X: real] :
( ( ord_less_eq_real @ ( a_seg @ X ) @ a )
= ( ord_less_eq_real @ X @ ( semiri5074537144036343181t_real @ n ) ) ) ).
% a_seg_le_a
thf(fact_71_zero__reorient,axiom,
! [X: real] :
( ( zero_zero_real = X )
= ( X = zero_zero_real ) ) ).
% zero_reorient
thf(fact_72_zero__reorient,axiom,
! [X: nat] :
( ( zero_zero_nat = X )
= ( X = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_73_zero__reorient,axiom,
! [X: int] :
( ( zero_zero_int = X )
= ( X = zero_zero_int ) ) ).
% zero_reorient
thf(fact_74_mult_Oleft__commute,axiom,
! [B: real,A2: real,C: real] :
( ( times_times_real @ B @ ( times_times_real @ A2 @ C ) )
= ( times_times_real @ A2 @ ( times_times_real @ B @ C ) ) ) ).
% mult.left_commute
thf(fact_75_mult_Oleft__commute,axiom,
! [B: nat,A2: nat,C: nat] :
( ( times_times_nat @ B @ ( times_times_nat @ A2 @ C ) )
= ( times_times_nat @ A2 @ ( times_times_nat @ B @ C ) ) ) ).
% mult.left_commute
thf(fact_76_mult_Oleft__commute,axiom,
! [B: int,A2: int,C: int] :
( ( times_times_int @ B @ ( times_times_int @ A2 @ C ) )
= ( times_times_int @ A2 @ ( times_times_int @ B @ C ) ) ) ).
% mult.left_commute
thf(fact_77_mult_Ocommute,axiom,
( times_times_real
= ( ^ [A3: real,B2: real] : ( times_times_real @ B2 @ A3 ) ) ) ).
% mult.commute
thf(fact_78_mult_Ocommute,axiom,
( times_times_nat
= ( ^ [A3: nat,B2: nat] : ( times_times_nat @ B2 @ A3 ) ) ) ).
% mult.commute
thf(fact_79_mult_Ocommute,axiom,
( times_times_int
= ( ^ [A3: int,B2: int] : ( times_times_int @ B2 @ A3 ) ) ) ).
% mult.commute
thf(fact_80_mult_Oassoc,axiom,
! [A2: real,B: real,C: real] :
( ( times_times_real @ ( times_times_real @ A2 @ B ) @ C )
= ( times_times_real @ A2 @ ( times_times_real @ B @ C ) ) ) ).
% mult.assoc
thf(fact_81_mult_Oassoc,axiom,
! [A2: nat,B: nat,C: nat] :
( ( times_times_nat @ ( times_times_nat @ A2 @ B ) @ C )
= ( times_times_nat @ A2 @ ( times_times_nat @ B @ C ) ) ) ).
% mult.assoc
thf(fact_82_mult_Oassoc,axiom,
! [A2: int,B: int,C: int] :
( ( times_times_int @ ( times_times_int @ A2 @ B ) @ C )
= ( times_times_int @ A2 @ ( times_times_int @ B @ C ) ) ) ).
% mult.assoc
thf(fact_83_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A2: real,B: real,C: real] :
( ( times_times_real @ ( times_times_real @ A2 @ B ) @ C )
= ( times_times_real @ A2 @ ( times_times_real @ B @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_84_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A2: nat,B: nat,C: nat] :
( ( times_times_nat @ ( times_times_nat @ A2 @ B ) @ C )
= ( times_times_nat @ A2 @ ( times_times_nat @ B @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_85_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A2: int,B: int,C: int] :
( ( times_times_int @ ( times_times_int @ A2 @ B ) @ C )
= ( times_times_int @ A2 @ ( times_times_int @ B @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_86_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
! [A2: real,C: real,B: real] :
( ( minus_minus_real @ ( minus_minus_real @ A2 @ C ) @ B )
= ( minus_minus_real @ ( minus_minus_real @ A2 @ B ) @ C ) ) ).
% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_87_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
! [A2: nat,C: nat,B: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A2 @ C ) @ B )
= ( minus_minus_nat @ ( minus_minus_nat @ A2 @ B ) @ C ) ) ).
% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_88_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
! [A2: int,C: int,B: int] :
( ( minus_minus_int @ ( minus_minus_int @ A2 @ C ) @ B )
= ( minus_minus_int @ ( minus_minus_int @ A2 @ B ) @ C ) ) ).
% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_89_diff__eq__diff__eq,axiom,
! [A2: real,B: real,C: real,D: real] :
( ( ( minus_minus_real @ A2 @ B )
= ( minus_minus_real @ C @ D ) )
=> ( ( A2 = B )
= ( C = D ) ) ) ).
% diff_eq_diff_eq
thf(fact_90_diff__eq__diff__eq,axiom,
! [A2: int,B: int,C: int,D: int] :
( ( ( minus_minus_int @ A2 @ B )
= ( minus_minus_int @ C @ D ) )
=> ( ( A2 = B )
= ( C = D ) ) ) ).
% diff_eq_diff_eq
thf(fact_91_sum_Oreindex__bij__witness,axiom,
! [S: set_real,I2: set_real > real,J: real > set_real,T: set_set_real,H: set_real > real,G: real > real] :
( ! [A4: real] :
( ( member_real @ A4 @ S )
=> ( ( I2 @ ( J @ A4 ) )
= A4 ) )
=> ( ! [A4: real] :
( ( member_real @ A4 @ S )
=> ( member_set_real @ ( J @ A4 ) @ T ) )
=> ( ! [B3: set_real] :
( ( member_set_real @ B3 @ T )
=> ( ( J @ ( I2 @ B3 ) )
= B3 ) )
=> ( ! [B3: set_real] :
( ( member_set_real @ B3 @ T )
=> ( member_real @ ( I2 @ B3 ) @ S ) )
=> ( ! [A4: real] :
( ( member_real @ A4 @ S )
=> ( ( H @ ( J @ A4 ) )
= ( G @ A4 ) ) )
=> ( ( groups8097168146408367636l_real @ G @ S )
= ( groups8702937949983641418l_real @ H @ T ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_92_sum_Oreindex__bij__witness,axiom,
! [S: set_nat,I2: set_real > nat,J: nat > set_real,T: set_set_real,H: set_real > real,G: nat > real] :
( ! [A4: nat] :
( ( member_nat @ A4 @ S )
=> ( ( I2 @ ( J @ A4 ) )
= A4 ) )
=> ( ! [A4: nat] :
( ( member_nat @ A4 @ S )
=> ( member_set_real @ ( J @ A4 ) @ T ) )
=> ( ! [B3: set_real] :
( ( member_set_real @ B3 @ T )
=> ( ( J @ ( I2 @ B3 ) )
= B3 ) )
=> ( ! [B3: set_real] :
( ( member_set_real @ B3 @ T )
=> ( member_nat @ ( I2 @ B3 ) @ S ) )
=> ( ! [A4: nat] :
( ( member_nat @ A4 @ S )
=> ( ( H @ ( J @ A4 ) )
= ( G @ A4 ) ) )
=> ( ( groups6591440286371151544t_real @ G @ S )
= ( groups8702937949983641418l_real @ H @ T ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_93_sum_Oreindex__bij__witness,axiom,
! [S: set_o,I2: set_real > $o,J: $o > set_real,T: set_set_real,H: set_real > real,G: $o > real] :
( ! [A4: $o] :
( ( member_o @ A4 @ S )
=> ( ( I2 @ ( J @ A4 ) )
= A4 ) )
=> ( ! [A4: $o] :
( ( member_o @ A4 @ S )
=> ( member_set_real @ ( J @ A4 ) @ T ) )
=> ( ! [B3: set_real] :
( ( member_set_real @ B3 @ T )
=> ( ( J @ ( I2 @ B3 ) )
= B3 ) )
=> ( ! [B3: set_real] :
( ( member_set_real @ B3 @ T )
=> ( member_o @ ( I2 @ B3 ) @ S ) )
=> ( ! [A4: $o] :
( ( member_o @ A4 @ S )
=> ( ( H @ ( J @ A4 ) )
= ( G @ A4 ) ) )
=> ( ( groups8691415230153176458o_real @ G @ S )
= ( groups8702937949983641418l_real @ H @ T ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_94_sum_Oreindex__bij__witness,axiom,
! [S: set_set_real,I2: real > set_real,J: set_real > real,T: set_real,H: real > real,G: set_real > real] :
( ! [A4: set_real] :
( ( member_set_real @ A4 @ S )
=> ( ( I2 @ ( J @ A4 ) )
= A4 ) )
=> ( ! [A4: set_real] :
( ( member_set_real @ A4 @ S )
=> ( member_real @ ( J @ A4 ) @ T ) )
=> ( ! [B3: real] :
( ( member_real @ B3 @ T )
=> ( ( J @ ( I2 @ B3 ) )
= B3 ) )
=> ( ! [B3: real] :
( ( member_real @ B3 @ T )
=> ( member_set_real @ ( I2 @ B3 ) @ S ) )
=> ( ! [A4: set_real] :
( ( member_set_real @ A4 @ S )
=> ( ( H @ ( J @ A4 ) )
= ( G @ A4 ) ) )
=> ( ( groups8702937949983641418l_real @ G @ S )
= ( groups8097168146408367636l_real @ H @ T ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_95_sum_Oreindex__bij__witness,axiom,
! [S: set_set_real,I2: nat > set_real,J: set_real > nat,T: set_nat,H: nat > real,G: set_real > real] :
( ! [A4: set_real] :
( ( member_set_real @ A4 @ S )
=> ( ( I2 @ ( J @ A4 ) )
= A4 ) )
=> ( ! [A4: set_real] :
( ( member_set_real @ A4 @ S )
=> ( member_nat @ ( J @ A4 ) @ T ) )
=> ( ! [B3: nat] :
( ( member_nat @ B3 @ T )
=> ( ( J @ ( I2 @ B3 ) )
= B3 ) )
=> ( ! [B3: nat] :
( ( member_nat @ B3 @ T )
=> ( member_set_real @ ( I2 @ B3 ) @ S ) )
=> ( ! [A4: set_real] :
( ( member_set_real @ A4 @ S )
=> ( ( H @ ( J @ A4 ) )
= ( G @ A4 ) ) )
=> ( ( groups8702937949983641418l_real @ G @ S )
= ( groups6591440286371151544t_real @ H @ T ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_96_sum_Oreindex__bij__witness,axiom,
! [S: set_set_real,I2: $o > set_real,J: set_real > $o,T: set_o,H: $o > real,G: set_real > real] :
( ! [A4: set_real] :
( ( member_set_real @ A4 @ S )
=> ( ( I2 @ ( J @ A4 ) )
= A4 ) )
=> ( ! [A4: set_real] :
( ( member_set_real @ A4 @ S )
=> ( member_o @ ( J @ A4 ) @ T ) )
=> ( ! [B3: $o] :
( ( member_o @ B3 @ T )
=> ( ( J @ ( I2 @ B3 ) )
= B3 ) )
=> ( ! [B3: $o] :
( ( member_o @ B3 @ T )
=> ( member_set_real @ ( I2 @ B3 ) @ S ) )
=> ( ! [A4: set_real] :
( ( member_set_real @ A4 @ S )
=> ( ( H @ ( J @ A4 ) )
= ( G @ A4 ) ) )
=> ( ( groups8702937949983641418l_real @ G @ S )
= ( groups8691415230153176458o_real @ H @ T ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_97_sum_Oreindex__bij__witness,axiom,
! [S: set_set_real,I2: set_real > set_real,J: set_real > set_real,T: set_set_real,H: set_real > real,G: set_real > real] :
( ! [A4: set_real] :
( ( member_set_real @ A4 @ S )
=> ( ( I2 @ ( J @ A4 ) )
= A4 ) )
=> ( ! [A4: set_real] :
( ( member_set_real @ A4 @ S )
=> ( member_set_real @ ( J @ A4 ) @ T ) )
=> ( ! [B3: set_real] :
( ( member_set_real @ B3 @ T )
=> ( ( J @ ( I2 @ B3 ) )
= B3 ) )
=> ( ! [B3: set_real] :
( ( member_set_real @ B3 @ T )
=> ( member_set_real @ ( I2 @ B3 ) @ S ) )
=> ( ! [A4: set_real] :
( ( member_set_real @ A4 @ S )
=> ( ( H @ ( J @ A4 ) )
= ( G @ A4 ) ) )
=> ( ( groups8702937949983641418l_real @ G @ S )
= ( groups8702937949983641418l_real @ H @ T ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_98_sum_Oeq__general__inverses,axiom,
! [B4: set_set_real,K: set_real > real,A: set_real,H: real > set_real,Gamma: set_real > real,Phi: real > real] :
( ! [Y2: set_real] :
( ( member_set_real @ Y2 @ B4 )
=> ( ( member_real @ ( K @ Y2 ) @ A )
& ( ( H @ ( K @ Y2 ) )
= Y2 ) ) )
=> ( ! [X3: real] :
( ( member_real @ X3 @ A )
=> ( ( member_set_real @ ( H @ X3 ) @ B4 )
& ( ( K @ ( H @ X3 ) )
= X3 )
& ( ( Gamma @ ( H @ X3 ) )
= ( Phi @ X3 ) ) ) )
=> ( ( groups8097168146408367636l_real @ Phi @ A )
= ( groups8702937949983641418l_real @ Gamma @ B4 ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_99_sum_Oeq__general__inverses,axiom,
! [B4: set_set_real,K: set_real > nat,A: set_nat,H: nat > set_real,Gamma: set_real > real,Phi: nat > real] :
( ! [Y2: set_real] :
( ( member_set_real @ Y2 @ B4 )
=> ( ( member_nat @ ( K @ Y2 ) @ A )
& ( ( H @ ( K @ Y2 ) )
= Y2 ) ) )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( ( member_set_real @ ( H @ X3 ) @ B4 )
& ( ( K @ ( H @ X3 ) )
= X3 )
& ( ( Gamma @ ( H @ X3 ) )
= ( Phi @ X3 ) ) ) )
=> ( ( groups6591440286371151544t_real @ Phi @ A )
= ( groups8702937949983641418l_real @ Gamma @ B4 ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_100_sum_Oeq__general__inverses,axiom,
! [B4: set_set_real,K: set_real > $o,A: set_o,H: $o > set_real,Gamma: set_real > real,Phi: $o > real] :
( ! [Y2: set_real] :
( ( member_set_real @ Y2 @ B4 )
=> ( ( member_o @ ( K @ Y2 ) @ A )
& ( ( H @ ( K @ Y2 ) )
= Y2 ) ) )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ A )
=> ( ( member_set_real @ ( H @ X3 ) @ B4 )
& ( ( K @ ( H @ X3 ) )
= X3 )
& ( ( Gamma @ ( H @ X3 ) )
= ( Phi @ X3 ) ) ) )
=> ( ( groups8691415230153176458o_real @ Phi @ A )
= ( groups8702937949983641418l_real @ Gamma @ B4 ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_101_sum_Oeq__general__inverses,axiom,
! [B4: set_real,K: real > set_real,A: set_set_real,H: set_real > real,Gamma: real > real,Phi: set_real > real] :
( ! [Y2: real] :
( ( member_real @ Y2 @ B4 )
=> ( ( member_set_real @ ( K @ Y2 ) @ A )
& ( ( H @ ( K @ Y2 ) )
= Y2 ) ) )
=> ( ! [X3: set_real] :
( ( member_set_real @ X3 @ A )
=> ( ( member_real @ ( H @ X3 ) @ B4 )
& ( ( K @ ( H @ X3 ) )
= X3 )
& ( ( Gamma @ ( H @ X3 ) )
= ( Phi @ X3 ) ) ) )
=> ( ( groups8702937949983641418l_real @ Phi @ A )
= ( groups8097168146408367636l_real @ Gamma @ B4 ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_102_sum_Oeq__general__inverses,axiom,
! [B4: set_nat,K: nat > set_real,A: set_set_real,H: set_real > nat,Gamma: nat > real,Phi: set_real > real] :
( ! [Y2: nat] :
( ( member_nat @ Y2 @ B4 )
=> ( ( member_set_real @ ( K @ Y2 ) @ A )
& ( ( H @ ( K @ Y2 ) )
= Y2 ) ) )
=> ( ! [X3: set_real] :
( ( member_set_real @ X3 @ A )
=> ( ( member_nat @ ( H @ X3 ) @ B4 )
& ( ( K @ ( H @ X3 ) )
= X3 )
& ( ( Gamma @ ( H @ X3 ) )
= ( Phi @ X3 ) ) ) )
=> ( ( groups8702937949983641418l_real @ Phi @ A )
= ( groups6591440286371151544t_real @ Gamma @ B4 ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_103_sum_Oeq__general__inverses,axiom,
! [B4: set_o,K: $o > set_real,A: set_set_real,H: set_real > $o,Gamma: $o > real,Phi: set_real > real] :
( ! [Y2: $o] :
( ( member_o @ Y2 @ B4 )
=> ( ( member_set_real @ ( K @ Y2 ) @ A )
& ( ( H @ ( K @ Y2 ) )
= Y2 ) ) )
=> ( ! [X3: set_real] :
( ( member_set_real @ X3 @ A )
=> ( ( member_o @ ( H @ X3 ) @ B4 )
& ( ( K @ ( H @ X3 ) )
= X3 )
& ( ( Gamma @ ( H @ X3 ) )
= ( Phi @ X3 ) ) ) )
=> ( ( groups8702937949983641418l_real @ Phi @ A )
= ( groups8691415230153176458o_real @ Gamma @ B4 ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_104_sum_Oeq__general__inverses,axiom,
! [B4: set_set_real,K: set_real > set_real,A: set_set_real,H: set_real > set_real,Gamma: set_real > real,Phi: set_real > real] :
( ! [Y2: set_real] :
( ( member_set_real @ Y2 @ B4 )
=> ( ( member_set_real @ ( K @ Y2 ) @ A )
& ( ( H @ ( K @ Y2 ) )
= Y2 ) ) )
=> ( ! [X3: set_real] :
( ( member_set_real @ X3 @ A )
=> ( ( member_set_real @ ( H @ X3 ) @ B4 )
& ( ( K @ ( H @ X3 ) )
= X3 )
& ( ( Gamma @ ( H @ X3 ) )
= ( Phi @ X3 ) ) ) )
=> ( ( groups8702937949983641418l_real @ Phi @ A )
= ( groups8702937949983641418l_real @ Gamma @ B4 ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_105_sum_Oeq__general,axiom,
! [B4: set_set_real,A: set_real,H: real > set_real,Gamma: set_real > real,Phi: real > real] :
( ! [Y2: set_real] :
( ( member_set_real @ Y2 @ B4 )
=> ? [X4: real] :
( ( member_real @ X4 @ A )
& ( ( H @ X4 )
= Y2 )
& ! [Ya: real] :
( ( ( member_real @ Ya @ A )
& ( ( H @ Ya )
= Y2 ) )
=> ( Ya = X4 ) ) ) )
=> ( ! [X3: real] :
( ( member_real @ X3 @ A )
=> ( ( member_set_real @ ( H @ X3 ) @ B4 )
& ( ( Gamma @ ( H @ X3 ) )
= ( Phi @ X3 ) ) ) )
=> ( ( groups8097168146408367636l_real @ Phi @ A )
= ( groups8702937949983641418l_real @ Gamma @ B4 ) ) ) ) ).
% sum.eq_general
thf(fact_106_sum_Oeq__general,axiom,
! [B4: set_set_real,A: set_nat,H: nat > set_real,Gamma: set_real > real,Phi: nat > real] :
( ! [Y2: set_real] :
( ( member_set_real @ Y2 @ B4 )
=> ? [X4: nat] :
( ( member_nat @ X4 @ A )
& ( ( H @ X4 )
= Y2 )
& ! [Ya: nat] :
( ( ( member_nat @ Ya @ A )
& ( ( H @ Ya )
= Y2 ) )
=> ( Ya = X4 ) ) ) )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( ( member_set_real @ ( H @ X3 ) @ B4 )
& ( ( Gamma @ ( H @ X3 ) )
= ( Phi @ X3 ) ) ) )
=> ( ( groups6591440286371151544t_real @ Phi @ A )
= ( groups8702937949983641418l_real @ Gamma @ B4 ) ) ) ) ).
% sum.eq_general
thf(fact_107_sum_Oeq__general,axiom,
! [B4: set_set_real,A: set_o,H: $o > set_real,Gamma: set_real > real,Phi: $o > real] :
( ! [Y2: set_real] :
( ( member_set_real @ Y2 @ B4 )
=> ? [X4: $o] :
( ( member_o @ X4 @ A )
& ( ( H @ X4 )
= Y2 )
& ! [Ya: $o] :
( ( ( member_o @ Ya @ A )
& ( ( H @ Ya )
= Y2 ) )
=> ( Ya = X4 ) ) ) )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ A )
=> ( ( member_set_real @ ( H @ X3 ) @ B4 )
& ( ( Gamma @ ( H @ X3 ) )
= ( Phi @ X3 ) ) ) )
=> ( ( groups8691415230153176458o_real @ Phi @ A )
= ( groups8702937949983641418l_real @ Gamma @ B4 ) ) ) ) ).
% sum.eq_general
thf(fact_108_sum_Oeq__general,axiom,
! [B4: set_real,A: set_set_real,H: set_real > real,Gamma: real > real,Phi: set_real > real] :
( ! [Y2: real] :
( ( member_real @ Y2 @ B4 )
=> ? [X4: set_real] :
( ( member_set_real @ X4 @ A )
& ( ( H @ X4 )
= Y2 )
& ! [Ya: set_real] :
( ( ( member_set_real @ Ya @ A )
& ( ( H @ Ya )
= Y2 ) )
=> ( Ya = X4 ) ) ) )
=> ( ! [X3: set_real] :
( ( member_set_real @ X3 @ A )
=> ( ( member_real @ ( H @ X3 ) @ B4 )
& ( ( Gamma @ ( H @ X3 ) )
= ( Phi @ X3 ) ) ) )
=> ( ( groups8702937949983641418l_real @ Phi @ A )
= ( groups8097168146408367636l_real @ Gamma @ B4 ) ) ) ) ).
% sum.eq_general
thf(fact_109_sum_Oeq__general,axiom,
! [B4: set_nat,A: set_set_real,H: set_real > nat,Gamma: nat > real,Phi: set_real > real] :
( ! [Y2: nat] :
( ( member_nat @ Y2 @ B4 )
=> ? [X4: set_real] :
( ( member_set_real @ X4 @ A )
& ( ( H @ X4 )
= Y2 )
& ! [Ya: set_real] :
( ( ( member_set_real @ Ya @ A )
& ( ( H @ Ya )
= Y2 ) )
=> ( Ya = X4 ) ) ) )
=> ( ! [X3: set_real] :
( ( member_set_real @ X3 @ A )
=> ( ( member_nat @ ( H @ X3 ) @ B4 )
& ( ( Gamma @ ( H @ X3 ) )
= ( Phi @ X3 ) ) ) )
=> ( ( groups8702937949983641418l_real @ Phi @ A )
= ( groups6591440286371151544t_real @ Gamma @ B4 ) ) ) ) ).
% sum.eq_general
thf(fact_110_sum_Oeq__general,axiom,
! [B4: set_o,A: set_set_real,H: set_real > $o,Gamma: $o > real,Phi: set_real > real] :
( ! [Y2: $o] :
( ( member_o @ Y2 @ B4 )
=> ? [X4: set_real] :
( ( member_set_real @ X4 @ A )
& ( ( H @ X4 )
= Y2 )
& ! [Ya: set_real] :
( ( ( member_set_real @ Ya @ A )
& ( ( H @ Ya )
= Y2 ) )
=> ( Ya = X4 ) ) ) )
=> ( ! [X3: set_real] :
( ( member_set_real @ X3 @ A )
=> ( ( member_o @ ( H @ X3 ) @ B4 )
& ( ( Gamma @ ( H @ X3 ) )
= ( Phi @ X3 ) ) ) )
=> ( ( groups8702937949983641418l_real @ Phi @ A )
= ( groups8691415230153176458o_real @ Gamma @ B4 ) ) ) ) ).
% sum.eq_general
thf(fact_111_sum_Oeq__general,axiom,
! [B4: set_set_real,A: set_set_real,H: set_real > set_real,Gamma: set_real > real,Phi: set_real > real] :
( ! [Y2: set_real] :
( ( member_set_real @ Y2 @ B4 )
=> ? [X4: set_real] :
( ( member_set_real @ X4 @ A )
& ( ( H @ X4 )
= Y2 )
& ! [Ya: set_real] :
( ( ( member_set_real @ Ya @ A )
& ( ( H @ Ya )
= Y2 ) )
=> ( Ya = X4 ) ) ) )
=> ( ! [X3: set_real] :
( ( member_set_real @ X3 @ A )
=> ( ( member_set_real @ ( H @ X3 ) @ B4 )
& ( ( Gamma @ ( H @ X3 ) )
= ( Phi @ X3 ) ) ) )
=> ( ( groups8702937949983641418l_real @ Phi @ A )
= ( groups8702937949983641418l_real @ Gamma @ B4 ) ) ) ) ).
% sum.eq_general
thf(fact_112_sum_Ocong,axiom,
! [A: set_set_real,B4: set_set_real,G: set_real > real,H: set_real > real] :
( ( A = B4 )
=> ( ! [X3: set_real] :
( ( member_set_real @ X3 @ B4 )
=> ( ( G @ X3 )
= ( H @ X3 ) ) )
=> ( ( groups8702937949983641418l_real @ G @ A )
= ( groups8702937949983641418l_real @ H @ B4 ) ) ) ) ).
% sum.cong
thf(fact_113_sum_Oswap,axiom,
! [G: set_real > set_real > real,B4: set_set_real,A: set_set_real] :
( ( groups8702937949983641418l_real
@ ^ [I: set_real] : ( groups8702937949983641418l_real @ ( G @ I ) @ B4 )
@ A )
= ( groups8702937949983641418l_real
@ ^ [J2: set_real] :
( groups8702937949983641418l_real
@ ^ [I: set_real] : ( G @ I @ J2 )
@ A )
@ B4 ) ) ).
% sum.swap
thf(fact_114_zero__le,axiom,
! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% zero_le
thf(fact_115_diff__eq__diff__less__eq,axiom,
! [A2: real,B: real,C: real,D: real] :
( ( ( minus_minus_real @ A2 @ B )
= ( minus_minus_real @ C @ D ) )
=> ( ( ord_less_eq_real @ A2 @ B )
= ( ord_less_eq_real @ C @ D ) ) ) ).
% diff_eq_diff_less_eq
thf(fact_116_diff__eq__diff__less__eq,axiom,
! [A2: int,B: int,C: int,D: int] :
( ( ( minus_minus_int @ A2 @ B )
= ( minus_minus_int @ C @ D ) )
=> ( ( ord_less_eq_int @ A2 @ B )
= ( ord_less_eq_int @ C @ D ) ) ) ).
% diff_eq_diff_less_eq
thf(fact_117_diff__right__mono,axiom,
! [A2: real,B: real,C: real] :
( ( ord_less_eq_real @ A2 @ B )
=> ( ord_less_eq_real @ ( minus_minus_real @ A2 @ C ) @ ( minus_minus_real @ B @ C ) ) ) ).
% diff_right_mono
thf(fact_118_diff__right__mono,axiom,
! [A2: int,B: int,C: int] :
( ( ord_less_eq_int @ A2 @ B )
=> ( ord_less_eq_int @ ( minus_minus_int @ A2 @ C ) @ ( minus_minus_int @ B @ C ) ) ) ).
% diff_right_mono
thf(fact_119_diff__left__mono,axiom,
! [B: real,A2: real,C: real] :
( ( ord_less_eq_real @ B @ A2 )
=> ( ord_less_eq_real @ ( minus_minus_real @ C @ A2 ) @ ( minus_minus_real @ C @ B ) ) ) ).
% diff_left_mono
thf(fact_120_diff__left__mono,axiom,
! [B: int,A2: int,C: int] :
( ( ord_less_eq_int @ B @ A2 )
=> ( ord_less_eq_int @ ( minus_minus_int @ C @ A2 ) @ ( minus_minus_int @ C @ B ) ) ) ).
% diff_left_mono
thf(fact_121_diff__mono,axiom,
! [A2: real,B: real,D: real,C: real] :
( ( ord_less_eq_real @ A2 @ B )
=> ( ( ord_less_eq_real @ D @ C )
=> ( ord_less_eq_real @ ( minus_minus_real @ A2 @ C ) @ ( minus_minus_real @ B @ D ) ) ) ) ).
% diff_mono
thf(fact_122_diff__mono,axiom,
! [A2: int,B: int,D: int,C: int] :
( ( ord_less_eq_int @ A2 @ B )
=> ( ( ord_less_eq_int @ D @ C )
=> ( ord_less_eq_int @ ( minus_minus_int @ A2 @ C ) @ ( minus_minus_int @ B @ D ) ) ) ) ).
% diff_mono
thf(fact_123_eq__iff__diff__eq__0,axiom,
( ( ^ [Y3: real,Z: real] : ( Y3 = Z ) )
= ( ^ [A3: real,B2: real] :
( ( minus_minus_real @ A3 @ B2 )
= zero_zero_real ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_124_eq__iff__diff__eq__0,axiom,
( ( ^ [Y3: int,Z: int] : ( Y3 = Z ) )
= ( ^ [A3: int,B2: int] :
( ( minus_minus_int @ A3 @ B2 )
= zero_zero_int ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_125_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: real > real,A: set_real] :
( ( ( groups8097168146408367636l_real @ G @ A )
!= zero_zero_real )
=> ~ ! [A4: real] :
( ( member_real @ A4 @ A )
=> ( ( G @ A4 )
= zero_zero_real ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_126_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: nat > real,A: set_nat] :
( ( ( groups6591440286371151544t_real @ G @ A )
!= zero_zero_real )
=> ~ ! [A4: nat] :
( ( member_nat @ A4 @ A )
=> ( ( G @ A4 )
= zero_zero_real ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_127_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: $o > real,A: set_o] :
( ( ( groups8691415230153176458o_real @ G @ A )
!= zero_zero_real )
=> ~ ! [A4: $o] :
( ( member_o @ A4 @ A )
=> ( ( G @ A4 )
= zero_zero_real ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_128_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: real > nat,A: set_real] :
( ( ( groups1935376822645274424al_nat @ G @ A )
!= zero_zero_nat )
=> ~ ! [A4: real] :
( ( member_real @ A4 @ A )
=> ( ( G @ A4 )
= zero_zero_nat ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_129_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: nat > nat,A: set_nat] :
( ( ( groups3542108847815614940at_nat @ G @ A )
!= zero_zero_nat )
=> ~ ! [A4: nat] :
( ( member_nat @ A4 @ A )
=> ( ( G @ A4 )
= zero_zero_nat ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_130_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: $o > nat,A: set_o] :
( ( ( groups8507830703676809646_o_nat @ G @ A )
!= zero_zero_nat )
=> ~ ! [A4: $o] :
( ( member_o @ A4 @ A )
=> ( ( G @ A4 )
= zero_zero_nat ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_131_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: real > int,A: set_real] :
( ( ( groups1932886352136224148al_int @ G @ A )
!= zero_zero_int )
=> ~ ! [A4: real] :
( ( member_real @ A4 @ A )
=> ( ( G @ A4 )
= zero_zero_int ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_132_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: nat > int,A: set_nat] :
( ( ( groups3539618377306564664at_int @ G @ A )
!= zero_zero_int )
=> ~ ! [A4: nat] :
( ( member_nat @ A4 @ A )
=> ( ( G @ A4 )
= zero_zero_int ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_133_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: $o > int,A: set_o] :
( ( ( groups8505340233167759370_o_int @ G @ A )
!= zero_zero_int )
=> ~ ! [A4: $o] :
( ( member_o @ A4 @ A )
=> ( ( G @ A4 )
= zero_zero_int ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_134_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: set_real > nat,A: set_set_real] :
( ( ( groups3012202523422989166al_nat @ G @ A )
!= zero_zero_nat )
=> ~ ! [A4: set_real] :
( ( member_set_real @ A4 @ A )
=> ( ( G @ A4 )
= zero_zero_nat ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_135_sum_Oneutral,axiom,
! [A: set_set_real,G: set_real > real] :
( ! [X3: set_real] :
( ( member_set_real @ X3 @ A )
=> ( ( G @ X3 )
= zero_zero_real ) )
=> ( ( groups8702937949983641418l_real @ G @ A )
= zero_zero_real ) ) ).
% sum.neutral
thf(fact_136_divide__divide__eq__left_H,axiom,
! [A2: real,B: real,C: real] :
( ( divide_divide_real @ ( divide_divide_real @ A2 @ B ) @ C )
= ( divide_divide_real @ A2 @ ( times_times_real @ C @ B ) ) ) ).
% divide_divide_eq_left'
thf(fact_137_divide__divide__times__eq,axiom,
! [X: real,Y: real,Z2: real,W: real] :
( ( divide_divide_real @ ( divide_divide_real @ X @ Y ) @ ( divide_divide_real @ Z2 @ W ) )
= ( divide_divide_real @ ( times_times_real @ X @ W ) @ ( times_times_real @ Y @ Z2 ) ) ) ).
% divide_divide_times_eq
thf(fact_138_times__divide__times__eq,axiom,
! [X: real,Y: real,Z2: real,W: real] :
( ( times_times_real @ ( divide_divide_real @ X @ Y ) @ ( divide_divide_real @ Z2 @ W ) )
= ( divide_divide_real @ ( times_times_real @ X @ Z2 ) @ ( times_times_real @ Y @ W ) ) ) ).
% times_divide_times_eq
thf(fact_139_diff__divide__distrib,axiom,
! [A2: real,B: real,C: real] :
( ( divide_divide_real @ ( minus_minus_real @ A2 @ B ) @ C )
= ( minus_minus_real @ ( divide_divide_real @ A2 @ C ) @ ( divide_divide_real @ B @ C ) ) ) ).
% diff_divide_distrib
thf(fact_140_abs__ge__self,axiom,
! [A2: real] : ( ord_less_eq_real @ A2 @ ( abs_abs_real @ A2 ) ) ).
% abs_ge_self
thf(fact_141_abs__ge__self,axiom,
! [A2: int] : ( ord_less_eq_int @ A2 @ ( abs_abs_int @ A2 ) ) ).
% abs_ge_self
thf(fact_142_abs__le__D1,axiom,
! [A2: real,B: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ A2 ) @ B )
=> ( ord_less_eq_real @ A2 @ B ) ) ).
% abs_le_D1
thf(fact_143_abs__le__D1,axiom,
! [A2: int,B: int] :
( ( ord_less_eq_int @ ( abs_abs_int @ A2 ) @ B )
=> ( ord_less_eq_int @ A2 @ B ) ) ).
% abs_le_D1
thf(fact_144_abs__minus__commute,axiom,
! [A2: real,B: real] :
( ( abs_abs_real @ ( minus_minus_real @ A2 @ B ) )
= ( abs_abs_real @ ( minus_minus_real @ B @ A2 ) ) ) ).
% abs_minus_commute
thf(fact_145_abs__minus__commute,axiom,
! [A2: int,B: int] :
( ( abs_abs_int @ ( minus_minus_int @ A2 @ B ) )
= ( abs_abs_int @ ( minus_minus_int @ B @ A2 ) ) ) ).
% abs_minus_commute
thf(fact_146_sum__mono,axiom,
! [K2: set_real,F: real > real,G: real > real] :
( ! [I3: real] :
( ( member_real @ I3 @ K2 )
=> ( ord_less_eq_real @ ( F @ I3 ) @ ( G @ I3 ) ) )
=> ( ord_less_eq_real @ ( groups8097168146408367636l_real @ F @ K2 ) @ ( groups8097168146408367636l_real @ G @ K2 ) ) ) ).
% sum_mono
thf(fact_147_sum__mono,axiom,
! [K2: set_nat,F: nat > real,G: nat > real] :
( ! [I3: nat] :
( ( member_nat @ I3 @ K2 )
=> ( ord_less_eq_real @ ( F @ I3 ) @ ( G @ I3 ) ) )
=> ( ord_less_eq_real @ ( groups6591440286371151544t_real @ F @ K2 ) @ ( groups6591440286371151544t_real @ G @ K2 ) ) ) ).
% sum_mono
thf(fact_148_sum__mono,axiom,
! [K2: set_o,F: $o > real,G: $o > real] :
( ! [I3: $o] :
( ( member_o @ I3 @ K2 )
=> ( ord_less_eq_real @ ( F @ I3 ) @ ( G @ I3 ) ) )
=> ( ord_less_eq_real @ ( groups8691415230153176458o_real @ F @ K2 ) @ ( groups8691415230153176458o_real @ G @ K2 ) ) ) ).
% sum_mono
thf(fact_149_sum__mono,axiom,
! [K2: set_real,F: real > nat,G: real > nat] :
( ! [I3: real] :
( ( member_real @ I3 @ K2 )
=> ( ord_less_eq_nat @ ( F @ I3 ) @ ( G @ I3 ) ) )
=> ( ord_less_eq_nat @ ( groups1935376822645274424al_nat @ F @ K2 ) @ ( groups1935376822645274424al_nat @ G @ K2 ) ) ) ).
% sum_mono
thf(fact_150_sum__mono,axiom,
! [K2: set_nat,F: nat > nat,G: nat > nat] :
( ! [I3: nat] :
( ( member_nat @ I3 @ K2 )
=> ( ord_less_eq_nat @ ( F @ I3 ) @ ( G @ I3 ) ) )
=> ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ F @ K2 ) @ ( groups3542108847815614940at_nat @ G @ K2 ) ) ) ).
% sum_mono
thf(fact_151_sum__mono,axiom,
! [K2: set_o,F: $o > nat,G: $o > nat] :
( ! [I3: $o] :
( ( member_o @ I3 @ K2 )
=> ( ord_less_eq_nat @ ( F @ I3 ) @ ( G @ I3 ) ) )
=> ( ord_less_eq_nat @ ( groups8507830703676809646_o_nat @ F @ K2 ) @ ( groups8507830703676809646_o_nat @ G @ K2 ) ) ) ).
% sum_mono
thf(fact_152_sum__mono,axiom,
! [K2: set_real,F: real > int,G: real > int] :
( ! [I3: real] :
( ( member_real @ I3 @ K2 )
=> ( ord_less_eq_int @ ( F @ I3 ) @ ( G @ I3 ) ) )
=> ( ord_less_eq_int @ ( groups1932886352136224148al_int @ F @ K2 ) @ ( groups1932886352136224148al_int @ G @ K2 ) ) ) ).
% sum_mono
thf(fact_153_sum__mono,axiom,
! [K2: set_nat,F: nat > int,G: nat > int] :
( ! [I3: nat] :
( ( member_nat @ I3 @ K2 )
=> ( ord_less_eq_int @ ( F @ I3 ) @ ( G @ I3 ) ) )
=> ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ K2 ) @ ( groups3539618377306564664at_int @ G @ K2 ) ) ) ).
% sum_mono
thf(fact_154_sum__mono,axiom,
! [K2: set_o,F: $o > int,G: $o > int] :
( ! [I3: $o] :
( ( member_o @ I3 @ K2 )
=> ( ord_less_eq_int @ ( F @ I3 ) @ ( G @ I3 ) ) )
=> ( ord_less_eq_int @ ( groups8505340233167759370_o_int @ F @ K2 ) @ ( groups8505340233167759370_o_int @ G @ K2 ) ) ) ).
% sum_mono
thf(fact_155_sum__mono,axiom,
! [K2: set_set_real,F: set_real > nat,G: set_real > nat] :
( ! [I3: set_real] :
( ( member_set_real @ I3 @ K2 )
=> ( ord_less_eq_nat @ ( F @ I3 ) @ ( G @ I3 ) ) )
=> ( ord_less_eq_nat @ ( groups3012202523422989166al_nat @ F @ K2 ) @ ( groups3012202523422989166al_nat @ G @ K2 ) ) ) ).
% sum_mono
thf(fact_156_sum__product,axiom,
! [F: set_real > real,A: set_set_real,G: set_real > real,B4: set_set_real] :
( ( times_times_real @ ( groups8702937949983641418l_real @ F @ A ) @ ( groups8702937949983641418l_real @ G @ B4 ) )
= ( groups8702937949983641418l_real
@ ^ [I: set_real] :
( groups8702937949983641418l_real
@ ^ [J2: set_real] : ( times_times_real @ ( F @ I ) @ ( G @ J2 ) )
@ B4 )
@ A ) ) ).
% sum_product
thf(fact_157_sum__distrib__right,axiom,
! [F: set_real > real,A: set_set_real,R: real] :
( ( times_times_real @ ( groups8702937949983641418l_real @ F @ A ) @ R )
= ( groups8702937949983641418l_real
@ ^ [N2: set_real] : ( times_times_real @ ( F @ N2 ) @ R )
@ A ) ) ).
% sum_distrib_right
thf(fact_158_sum__distrib__left,axiom,
! [R: real,F: set_real > real,A: set_set_real] :
( ( times_times_real @ R @ ( groups8702937949983641418l_real @ F @ A ) )
= ( groups8702937949983641418l_real
@ ^ [N2: set_real] : ( times_times_real @ R @ ( F @ N2 ) )
@ A ) ) ).
% sum_distrib_left
thf(fact_159_sum__subtractf,axiom,
! [F: set_real > real,G: set_real > real,A: set_set_real] :
( ( groups8702937949983641418l_real
@ ^ [X2: set_real] : ( minus_minus_real @ ( F @ X2 ) @ ( G @ X2 ) )
@ A )
= ( minus_minus_real @ ( groups8702937949983641418l_real @ F @ A ) @ ( groups8702937949983641418l_real @ G @ A ) ) ) ).
% sum_subtractf
thf(fact_160_sum__divide__distrib,axiom,
! [F: set_real > real,A: set_set_real,R: real] :
( ( divide_divide_real @ ( groups8702937949983641418l_real @ F @ A ) @ R )
= ( groups8702937949983641418l_real
@ ^ [N2: set_real] : ( divide_divide_real @ ( F @ N2 ) @ R )
@ A ) ) ).
% sum_divide_distrib
thf(fact_161_le__iff__diff__le__0,axiom,
( ord_less_eq_real
= ( ^ [A3: real,B2: real] : ( ord_less_eq_real @ ( minus_minus_real @ A3 @ B2 ) @ zero_zero_real ) ) ) ).
% le_iff_diff_le_0
thf(fact_162_le__iff__diff__le__0,axiom,
( ord_less_eq_int
= ( ^ [A3: int,B2: int] : ( ord_less_eq_int @ ( minus_minus_int @ A3 @ B2 ) @ zero_zero_int ) ) ) ).
% le_iff_diff_le_0
thf(fact_163_divide__right__mono__neg,axiom,
! [A2: real,B: real,C: real] :
( ( ord_less_eq_real @ A2 @ B )
=> ( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ ( divide_divide_real @ A2 @ C ) ) ) ) ).
% divide_right_mono_neg
thf(fact_164_divide__nonpos__nonpos,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ zero_zero_real )
=> ( ( ord_less_eq_real @ Y @ zero_zero_real )
=> ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X @ Y ) ) ) ) ).
% divide_nonpos_nonpos
thf(fact_165_divide__nonpos__nonneg,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ zero_zero_real )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ord_less_eq_real @ ( divide_divide_real @ X @ Y ) @ zero_zero_real ) ) ) ).
% divide_nonpos_nonneg
thf(fact_166_divide__nonneg__nonpos,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ Y @ zero_zero_real )
=> ( ord_less_eq_real @ ( divide_divide_real @ X @ Y ) @ zero_zero_real ) ) ) ).
% divide_nonneg_nonpos
thf(fact_167_divide__nonneg__nonneg,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X @ Y ) ) ) ) ).
% divide_nonneg_nonneg
thf(fact_168_zero__le__divide__iff,axiom,
! [A2: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ A2 @ B ) )
= ( ( ( ord_less_eq_real @ zero_zero_real @ A2 )
& ( ord_less_eq_real @ zero_zero_real @ B ) )
| ( ( ord_less_eq_real @ A2 @ zero_zero_real )
& ( ord_less_eq_real @ B @ zero_zero_real ) ) ) ) ).
% zero_le_divide_iff
thf(fact_169_divide__right__mono,axiom,
! [A2: real,B: real,C: real] :
( ( ord_less_eq_real @ A2 @ B )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ ( divide_divide_real @ A2 @ C ) @ ( divide_divide_real @ B @ C ) ) ) ) ).
% divide_right_mono
thf(fact_170_divide__le__0__iff,axiom,
! [A2: real,B: real] :
( ( ord_less_eq_real @ ( divide_divide_real @ A2 @ B ) @ zero_zero_real )
= ( ( ( ord_less_eq_real @ zero_zero_real @ A2 )
& ( ord_less_eq_real @ B @ zero_zero_real ) )
| ( ( ord_less_eq_real @ A2 @ zero_zero_real )
& ( ord_less_eq_real @ zero_zero_real @ B ) ) ) ) ).
% divide_le_0_iff
thf(fact_171_sum__nonpos,axiom,
! [A: set_real,F: real > real] :
( ! [X3: real] :
( ( member_real @ X3 @ A )
=> ( ord_less_eq_real @ ( F @ X3 ) @ zero_zero_real ) )
=> ( ord_less_eq_real @ ( groups8097168146408367636l_real @ F @ A ) @ zero_zero_real ) ) ).
% sum_nonpos
thf(fact_172_sum__nonpos,axiom,
! [A: set_nat,F: nat > real] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( ord_less_eq_real @ ( F @ X3 ) @ zero_zero_real ) )
=> ( ord_less_eq_real @ ( groups6591440286371151544t_real @ F @ A ) @ zero_zero_real ) ) ).
% sum_nonpos
thf(fact_173_sum__nonpos,axiom,
! [A: set_o,F: $o > real] :
( ! [X3: $o] :
( ( member_o @ X3 @ A )
=> ( ord_less_eq_real @ ( F @ X3 ) @ zero_zero_real ) )
=> ( ord_less_eq_real @ ( groups8691415230153176458o_real @ F @ A ) @ zero_zero_real ) ) ).
% sum_nonpos
thf(fact_174_sum__nonpos,axiom,
! [A: set_real,F: real > nat] :
( ! [X3: real] :
( ( member_real @ X3 @ A )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ zero_zero_nat ) )
=> ( ord_less_eq_nat @ ( groups1935376822645274424al_nat @ F @ A ) @ zero_zero_nat ) ) ).
% sum_nonpos
thf(fact_175_sum__nonpos,axiom,
! [A: set_nat,F: nat > nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ zero_zero_nat ) )
=> ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ F @ A ) @ zero_zero_nat ) ) ).
% sum_nonpos
thf(fact_176_sum__nonpos,axiom,
! [A: set_o,F: $o > nat] :
( ! [X3: $o] :
( ( member_o @ X3 @ A )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ zero_zero_nat ) )
=> ( ord_less_eq_nat @ ( groups8507830703676809646_o_nat @ F @ A ) @ zero_zero_nat ) ) ).
% sum_nonpos
thf(fact_177_sum__nonpos,axiom,
! [A: set_real,F: real > int] :
( ! [X3: real] :
( ( member_real @ X3 @ A )
=> ( ord_less_eq_int @ ( F @ X3 ) @ zero_zero_int ) )
=> ( ord_less_eq_int @ ( groups1932886352136224148al_int @ F @ A ) @ zero_zero_int ) ) ).
% sum_nonpos
thf(fact_178_sum__nonpos,axiom,
! [A: set_nat,F: nat > int] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( ord_less_eq_int @ ( F @ X3 ) @ zero_zero_int ) )
=> ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ A ) @ zero_zero_int ) ) ).
% sum_nonpos
thf(fact_179_sum__nonpos,axiom,
! [A: set_o,F: $o > int] :
( ! [X3: $o] :
( ( member_o @ X3 @ A )
=> ( ord_less_eq_int @ ( F @ X3 ) @ zero_zero_int ) )
=> ( ord_less_eq_int @ ( groups8505340233167759370_o_int @ F @ A ) @ zero_zero_int ) ) ).
% sum_nonpos
thf(fact_180_sum__nonpos,axiom,
! [A: set_set_real,F: set_real > nat] :
( ! [X3: set_real] :
( ( member_set_real @ X3 @ A )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ zero_zero_nat ) )
=> ( ord_less_eq_nat @ ( groups3012202523422989166al_nat @ F @ A ) @ zero_zero_nat ) ) ).
% sum_nonpos
thf(fact_181_sum__nonneg,axiom,
! [A: set_real,F: real > real] :
( ! [X3: real] :
( ( member_real @ X3 @ A )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) )
=> ( ord_less_eq_real @ zero_zero_real @ ( groups8097168146408367636l_real @ F @ A ) ) ) ).
% sum_nonneg
thf(fact_182_sum__nonneg,axiom,
! [A: set_nat,F: nat > real] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) )
=> ( ord_less_eq_real @ zero_zero_real @ ( groups6591440286371151544t_real @ F @ A ) ) ) ).
% sum_nonneg
thf(fact_183_sum__nonneg,axiom,
! [A: set_o,F: $o > real] :
( ! [X3: $o] :
( ( member_o @ X3 @ A )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) )
=> ( ord_less_eq_real @ zero_zero_real @ ( groups8691415230153176458o_real @ F @ A ) ) ) ).
% sum_nonneg
thf(fact_184_sum__nonneg,axiom,
! [A: set_real,F: real > nat] :
( ! [X3: real] :
( ( member_real @ X3 @ A )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X3 ) ) )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( groups1935376822645274424al_nat @ F @ A ) ) ) ).
% sum_nonneg
thf(fact_185_sum__nonneg,axiom,
! [A: set_nat,F: nat > nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X3 ) ) )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( groups3542108847815614940at_nat @ F @ A ) ) ) ).
% sum_nonneg
thf(fact_186_sum__nonneg,axiom,
! [A: set_o,F: $o > nat] :
( ! [X3: $o] :
( ( member_o @ X3 @ A )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X3 ) ) )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( groups8507830703676809646_o_nat @ F @ A ) ) ) ).
% sum_nonneg
thf(fact_187_sum__nonneg,axiom,
! [A: set_real,F: real > int] :
( ! [X3: real] :
( ( member_real @ X3 @ A )
=> ( ord_less_eq_int @ zero_zero_int @ ( F @ X3 ) ) )
=> ( ord_less_eq_int @ zero_zero_int @ ( groups1932886352136224148al_int @ F @ A ) ) ) ).
% sum_nonneg
thf(fact_188_sum__nonneg,axiom,
! [A: set_nat,F: nat > int] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( ord_less_eq_int @ zero_zero_int @ ( F @ X3 ) ) )
=> ( ord_less_eq_int @ zero_zero_int @ ( groups3539618377306564664at_int @ F @ A ) ) ) ).
% sum_nonneg
thf(fact_189_sum__nonneg,axiom,
! [A: set_o,F: $o > int] :
( ! [X3: $o] :
( ( member_o @ X3 @ A )
=> ( ord_less_eq_int @ zero_zero_int @ ( F @ X3 ) ) )
=> ( ord_less_eq_int @ zero_zero_int @ ( groups8505340233167759370_o_int @ F @ A ) ) ) ).
% sum_nonneg
thf(fact_190_sum__nonneg,axiom,
! [A: set_set_real,F: set_real > nat] :
( ! [X3: set_real] :
( ( member_set_real @ X3 @ A )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X3 ) ) )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( groups3012202523422989166al_nat @ F @ A ) ) ) ).
% sum_nonneg
thf(fact_191_nonzero__eq__divide__eq,axiom,
! [C: real,A2: real,B: real] :
( ( C != zero_zero_real )
=> ( ( A2
= ( divide_divide_real @ B @ C ) )
= ( ( times_times_real @ A2 @ C )
= B ) ) ) ).
% nonzero_eq_divide_eq
thf(fact_192_nonzero__divide__eq__eq,axiom,
! [C: real,B: real,A2: real] :
( ( C != zero_zero_real )
=> ( ( ( divide_divide_real @ B @ C )
= A2 )
= ( B
= ( times_times_real @ A2 @ C ) ) ) ) ).
% nonzero_divide_eq_eq
thf(fact_193_eq__divide__imp,axiom,
! [C: real,A2: real,B: real] :
( ( C != zero_zero_real )
=> ( ( ( times_times_real @ A2 @ C )
= B )
=> ( A2
= ( divide_divide_real @ B @ C ) ) ) ) ).
% eq_divide_imp
thf(fact_194_divide__eq__imp,axiom,
! [C: real,B: real,A2: real] :
( ( C != zero_zero_real )
=> ( ( B
= ( times_times_real @ A2 @ C ) )
=> ( ( divide_divide_real @ B @ C )
= A2 ) ) ) ).
% divide_eq_imp
thf(fact_195_eq__divide__eq,axiom,
! [A2: real,B: real,C: real] :
( ( A2
= ( divide_divide_real @ B @ C ) )
= ( ( ( C != zero_zero_real )
=> ( ( times_times_real @ A2 @ C )
= B ) )
& ( ( C = zero_zero_real )
=> ( A2 = zero_zero_real ) ) ) ) ).
% eq_divide_eq
thf(fact_196_divide__eq__eq,axiom,
! [B: real,C: real,A2: real] :
( ( ( divide_divide_real @ B @ C )
= A2 )
= ( ( ( C != zero_zero_real )
=> ( B
= ( times_times_real @ A2 @ C ) ) )
& ( ( C = zero_zero_real )
=> ( A2 = zero_zero_real ) ) ) ) ).
% divide_eq_eq
thf(fact_197_frac__eq__eq,axiom,
! [Y: real,Z2: real,X: real,W: real] :
( ( Y != zero_zero_real )
=> ( ( Z2 != zero_zero_real )
=> ( ( ( divide_divide_real @ X @ Y )
= ( divide_divide_real @ W @ Z2 ) )
= ( ( times_times_real @ X @ Z2 )
= ( times_times_real @ W @ Y ) ) ) ) ) ).
% frac_eq_eq
thf(fact_198_abs__ge__zero,axiom,
! [A2: real] : ( ord_less_eq_real @ zero_zero_real @ ( abs_abs_real @ A2 ) ) ).
% abs_ge_zero
thf(fact_199_abs__ge__zero,axiom,
! [A2: int] : ( ord_less_eq_int @ zero_zero_int @ ( abs_abs_int @ A2 ) ) ).
% abs_ge_zero
thf(fact_200_abs__triangle__ineq2__sym,axiom,
! [A2: real,B: real] : ( ord_less_eq_real @ ( minus_minus_real @ ( abs_abs_real @ A2 ) @ ( abs_abs_real @ B ) ) @ ( abs_abs_real @ ( minus_minus_real @ B @ A2 ) ) ) ).
% abs_triangle_ineq2_sym
thf(fact_201_abs__triangle__ineq2__sym,axiom,
! [A2: int,B: int] : ( ord_less_eq_int @ ( minus_minus_int @ ( abs_abs_int @ A2 ) @ ( abs_abs_int @ B ) ) @ ( abs_abs_int @ ( minus_minus_int @ B @ A2 ) ) ) ).
% abs_triangle_ineq2_sym
thf(fact_202_abs__triangle__ineq3,axiom,
! [A2: real,B: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( abs_abs_real @ A2 ) @ ( abs_abs_real @ B ) ) ) @ ( abs_abs_real @ ( minus_minus_real @ A2 @ B ) ) ) ).
% abs_triangle_ineq3
thf(fact_203_abs__triangle__ineq3,axiom,
! [A2: int,B: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( abs_abs_int @ A2 ) @ ( abs_abs_int @ B ) ) ) @ ( abs_abs_int @ ( minus_minus_int @ A2 @ B ) ) ) ).
% abs_triangle_ineq3
thf(fact_204_abs__triangle__ineq2,axiom,
! [A2: real,B: real] : ( ord_less_eq_real @ ( minus_minus_real @ ( abs_abs_real @ A2 ) @ ( abs_abs_real @ B ) ) @ ( abs_abs_real @ ( minus_minus_real @ A2 @ B ) ) ) ).
% abs_triangle_ineq2
thf(fact_205_abs__triangle__ineq2,axiom,
! [A2: int,B: int] : ( ord_less_eq_int @ ( minus_minus_int @ ( abs_abs_int @ A2 ) @ ( abs_abs_int @ B ) ) @ ( abs_abs_int @ ( minus_minus_int @ A2 @ B ) ) ) ).
% abs_triangle_ineq2
thf(fact_206_nonzero__abs__divide,axiom,
! [B: real,A2: real] :
( ( B != zero_zero_real )
=> ( ( abs_abs_real @ ( divide_divide_real @ A2 @ B ) )
= ( divide_divide_real @ ( abs_abs_real @ A2 ) @ ( abs_abs_real @ B ) ) ) ) ).
% nonzero_abs_divide
thf(fact_207_divide__diff__eq__iff,axiom,
! [Z2: real,X: real,Y: real] :
( ( Z2 != zero_zero_real )
=> ( ( minus_minus_real @ ( divide_divide_real @ X @ Z2 ) @ Y )
= ( divide_divide_real @ ( minus_minus_real @ X @ ( times_times_real @ Y @ Z2 ) ) @ Z2 ) ) ) ).
% divide_diff_eq_iff
thf(fact_208_diff__divide__eq__iff,axiom,
! [Z2: real,X: real,Y: real] :
( ( Z2 != zero_zero_real )
=> ( ( minus_minus_real @ X @ ( divide_divide_real @ Y @ Z2 ) )
= ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X @ Z2 ) @ Y ) @ Z2 ) ) ) ).
% diff_divide_eq_iff
thf(fact_209_diff__frac__eq,axiom,
! [Y: real,Z2: real,X: real,W: real] :
( ( Y != zero_zero_real )
=> ( ( Z2 != zero_zero_real )
=> ( ( minus_minus_real @ ( divide_divide_real @ X @ Y ) @ ( divide_divide_real @ W @ Z2 ) )
= ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X @ Z2 ) @ ( times_times_real @ W @ Y ) ) @ ( times_times_real @ Y @ Z2 ) ) ) ) ) ).
% diff_frac_eq
thf(fact_210_add__divide__eq__if__simps_I4_J,axiom,
! [Z2: real,A2: real,B: real] :
( ( ( Z2 = zero_zero_real )
=> ( ( minus_minus_real @ A2 @ ( divide_divide_real @ B @ Z2 ) )
= A2 ) )
& ( ( Z2 != zero_zero_real )
=> ( ( minus_minus_real @ A2 @ ( divide_divide_real @ B @ Z2 ) )
= ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ A2 @ Z2 ) @ B ) @ Z2 ) ) ) ) ).
% add_divide_eq_if_simps(4)
thf(fact_211_frac__le__eq,axiom,
! [Y: real,Z2: real,X: real,W: real] :
( ( Y != zero_zero_real )
=> ( ( Z2 != zero_zero_real )
=> ( ( ord_less_eq_real @ ( divide_divide_real @ X @ Y ) @ ( divide_divide_real @ W @ Z2 ) )
= ( ord_less_eq_real @ ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X @ Z2 ) @ ( times_times_real @ W @ Y ) ) @ ( times_times_real @ Y @ Z2 ) ) @ zero_zero_real ) ) ) ) ).
% frac_le_eq
thf(fact_212_int__21__D,axiom,
! [K2: set_real] :
( ( member_set_real @ K2 @ ( regular_division @ zero_zero_real @ a @ n ) )
=> ( hensto240673015341029504l_real
@ ^ [X2: real] : ( minus_minus_real @ ( f2 @ X2 ) @ ( f1 @ X2 ) )
@ ( times_times_real @ ( minus_minus_real @ ( f @ ( comple1385675409528146559p_real @ K2 ) ) @ ( f @ ( comple4887499456419720421f_real @ K2 ) ) ) @ ( divide_divide_real @ a @ ( semiri5074537144036343181t_real @ n ) ) )
@ K2 ) ) ).
% int_21_D
thf(fact_213_int__f1__D,axiom,
! [K2: set_real] :
( ( member_set_real @ K2 @ ( regular_division @ zero_zero_real @ a @ n ) )
=> ( hensto240673015341029504l_real @ f1 @ ( times_times_real @ ( f @ ( comple4887499456419720421f_real @ K2 ) ) @ ( divide_divide_real @ a @ ( semiri5074537144036343181t_real @ n ) ) ) @ K2 ) ) ).
% int_f1_D
thf(fact_214_int__f2__D,axiom,
! [K2: set_real] :
( ( member_set_real @ K2 @ ( regular_division @ zero_zero_real @ a @ n ) )
=> ( hensto240673015341029504l_real @ f2 @ ( times_times_real @ ( f @ ( comple1385675409528146559p_real @ K2 ) ) @ ( divide_divide_real @ a @ ( semiri5074537144036343181t_real @ n ) ) ) @ K2 ) ) ).
% int_f2_D
thf(fact_215_div__mult__mult1__if,axiom,
! [C: nat,A2: nat,B: nat] :
( ( ( C = zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ C @ A2 ) @ ( times_times_nat @ C @ B ) )
= zero_zero_nat ) )
& ( ( C != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ C @ A2 ) @ ( times_times_nat @ C @ B ) )
= ( divide_divide_nat @ A2 @ B ) ) ) ) ).
% div_mult_mult1_if
thf(fact_216_div__mult__mult1__if,axiom,
! [C: int,A2: int,B: int] :
( ( ( C = zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B ) )
= zero_zero_int ) )
& ( ( C != zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B ) )
= ( divide_divide_int @ A2 @ B ) ) ) ) ).
% div_mult_mult1_if
thf(fact_217_div__mult__mult2,axiom,
! [C: nat,A2: nat,B: nat] :
( ( C != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B @ C ) )
= ( divide_divide_nat @ A2 @ B ) ) ) ).
% div_mult_mult2
thf(fact_218_div__mult__mult2,axiom,
! [C: int,A2: int,B: int] :
( ( C != zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B @ C ) )
= ( divide_divide_int @ A2 @ B ) ) ) ).
% div_mult_mult2
thf(fact_219_div__mult__mult1,axiom,
! [C: nat,A2: nat,B: nat] :
( ( C != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ C @ A2 ) @ ( times_times_nat @ C @ B ) )
= ( divide_divide_nat @ A2 @ B ) ) ) ).
% div_mult_mult1
thf(fact_220_div__mult__mult1,axiom,
! [C: int,A2: int,B: int] :
( ( C != zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B ) )
= ( divide_divide_int @ A2 @ B ) ) ) ).
% div_mult_mult1
thf(fact_221_nonzero__mult__div__cancel__right,axiom,
! [B: real,A2: real] :
( ( B != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ A2 @ B ) @ B )
= A2 ) ) ).
% nonzero_mult_div_cancel_right
thf(fact_222_nonzero__mult__div__cancel__right,axiom,
! [B: nat,A2: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ A2 @ B ) @ B )
= A2 ) ) ).
% nonzero_mult_div_cancel_right
thf(fact_223_nonzero__mult__div__cancel__right,axiom,
! [B: int,A2: int] :
( ( B != zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ A2 @ B ) @ B )
= A2 ) ) ).
% nonzero_mult_div_cancel_right
thf(fact_224_nonzero__mult__div__cancel__left,axiom,
! [A2: real,B: real] :
( ( A2 != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ A2 @ B ) @ A2 )
= B ) ) ).
% nonzero_mult_div_cancel_left
thf(fact_225_nonzero__mult__div__cancel__left,axiom,
! [A2: nat,B: nat] :
( ( A2 != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ A2 @ B ) @ A2 )
= B ) ) ).
% nonzero_mult_div_cancel_left
thf(fact_226_nonzero__mult__div__cancel__left,axiom,
! [A2: int,B: int] :
( ( A2 != zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ A2 @ B ) @ A2 )
= B ) ) ).
% nonzero_mult_div_cancel_left
thf(fact_227_of__nat__le__0__iff,axiom,
! [M: nat] :
( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M ) @ zero_zero_real )
= ( M = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_228_of__nat__le__0__iff,axiom,
! [M: nat] :
( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M ) @ zero_zero_nat )
= ( M = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_229_of__nat__le__0__iff,axiom,
! [M: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ zero_zero_int )
= ( M = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_230_of__nat__sum,axiom,
! [F: set_real > nat,A: set_set_real] :
( ( semiri5074537144036343181t_real @ ( groups3012202523422989166al_nat @ F @ A ) )
= ( groups8702937949983641418l_real
@ ^ [X2: set_real] : ( semiri5074537144036343181t_real @ ( F @ X2 ) )
@ A ) ) ).
% of_nat_sum
thf(fact_231_D__ne,axiom,
( ( regular_division @ zero_zero_real @ a @ n )
!= bot_bot_set_set_real ) ).
% D_ne
thf(fact_232_bot__nat__0_Oextremum,axiom,
! [A2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A2 ) ).
% bot_nat_0.extremum
thf(fact_233_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_234_mult__is__0,axiom,
! [M: nat,N: nat] :
( ( ( times_times_nat @ M @ N )
= zero_zero_nat )
= ( ( M = zero_zero_nat )
| ( N = zero_zero_nat ) ) ) ).
% mult_is_0
thf(fact_235_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_236_diff__is__0__eq,axiom,
! [M: nat,N: nat] :
( ( ( minus_minus_nat @ M @ N )
= zero_zero_nat )
= ( ord_less_eq_nat @ M @ N ) ) ).
% diff_is_0_eq
thf(fact_237_mult__0__right,axiom,
! [M: nat] :
( ( times_times_nat @ M @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_0_right
thf(fact_238_mult__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N ) )
= ( ( M = N )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel1
thf(fact_239_mult__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ( times_times_nat @ M @ K )
= ( times_times_nat @ N @ K ) )
= ( ( M = N )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel2
thf(fact_240_diff__is__0__eq_H,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( minus_minus_nat @ M @ N )
= zero_zero_nat ) ) ).
% diff_is_0_eq'
thf(fact_241_diff__self__eq__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ M )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_242_of__nat__eq__iff,axiom,
! [M: nat,N: nat] :
( ( ( semiri5074537144036343181t_real @ M )
= ( semiri5074537144036343181t_real @ N ) )
= ( M = N ) ) ).
% of_nat_eq_iff
thf(fact_243_of__nat__eq__iff,axiom,
! [M: nat,N: nat] :
( ( ( semiri1314217659103216013at_int @ M )
= ( semiri1314217659103216013at_int @ N ) )
= ( M = N ) ) ).
% of_nat_eq_iff
thf(fact_244_abs__abs,axiom,
! [A2: real] :
( ( abs_abs_real @ ( abs_abs_real @ A2 ) )
= ( abs_abs_real @ A2 ) ) ).
% abs_abs
thf(fact_245_mult__zero__left,axiom,
! [A2: real] :
( ( times_times_real @ zero_zero_real @ A2 )
= zero_zero_real ) ).
% mult_zero_left
thf(fact_246_mult__zero__left,axiom,
! [A2: nat] :
( ( times_times_nat @ zero_zero_nat @ A2 )
= zero_zero_nat ) ).
% mult_zero_left
thf(fact_247_mult__zero__left,axiom,
! [A2: int] :
( ( times_times_int @ zero_zero_int @ A2 )
= zero_zero_int ) ).
% mult_zero_left
thf(fact_248_mult__zero__right,axiom,
! [A2: real] :
( ( times_times_real @ A2 @ zero_zero_real )
= zero_zero_real ) ).
% mult_zero_right
thf(fact_249_mult__zero__right,axiom,
! [A2: nat] :
( ( times_times_nat @ A2 @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_zero_right
thf(fact_250_mult__zero__right,axiom,
! [A2: int] :
( ( times_times_int @ A2 @ zero_zero_int )
= zero_zero_int ) ).
% mult_zero_right
thf(fact_251_mult__eq__0__iff,axiom,
! [A2: real,B: real] :
( ( ( times_times_real @ A2 @ B )
= zero_zero_real )
= ( ( A2 = zero_zero_real )
| ( B = zero_zero_real ) ) ) ).
% mult_eq_0_iff
thf(fact_252_mult__eq__0__iff,axiom,
! [A2: nat,B: nat] :
( ( ( times_times_nat @ A2 @ B )
= zero_zero_nat )
= ( ( A2 = zero_zero_nat )
| ( B = zero_zero_nat ) ) ) ).
% mult_eq_0_iff
thf(fact_253_mult__eq__0__iff,axiom,
! [A2: int,B: int] :
( ( ( times_times_int @ A2 @ B )
= zero_zero_int )
= ( ( A2 = zero_zero_int )
| ( B = zero_zero_int ) ) ) ).
% mult_eq_0_iff
thf(fact_254_mult__cancel__left,axiom,
! [C: real,A2: real,B: real] :
( ( ( times_times_real @ C @ A2 )
= ( times_times_real @ C @ B ) )
= ( ( C = zero_zero_real )
| ( A2 = B ) ) ) ).
% mult_cancel_left
thf(fact_255_mult__cancel__left,axiom,
! [C: nat,A2: nat,B: nat] :
( ( ( times_times_nat @ C @ A2 )
= ( times_times_nat @ C @ B ) )
= ( ( C = zero_zero_nat )
| ( A2 = B ) ) ) ).
% mult_cancel_left
thf(fact_256_mult__cancel__left,axiom,
! [C: int,A2: int,B: int] :
( ( ( times_times_int @ C @ A2 )
= ( times_times_int @ C @ B ) )
= ( ( C = zero_zero_int )
| ( A2 = B ) ) ) ).
% mult_cancel_left
thf(fact_257_mult__cancel__right,axiom,
! [A2: real,C: real,B: real] :
( ( ( times_times_real @ A2 @ C )
= ( times_times_real @ B @ C ) )
= ( ( C = zero_zero_real )
| ( A2 = B ) ) ) ).
% mult_cancel_right
thf(fact_258_mult__cancel__right,axiom,
! [A2: nat,C: nat,B: nat] :
( ( ( times_times_nat @ A2 @ C )
= ( times_times_nat @ B @ C ) )
= ( ( C = zero_zero_nat )
| ( A2 = B ) ) ) ).
% mult_cancel_right
thf(fact_259_mult__cancel__right,axiom,
! [A2: int,C: int,B: int] :
( ( ( times_times_int @ A2 @ C )
= ( times_times_int @ B @ C ) )
= ( ( C = zero_zero_int )
| ( A2 = B ) ) ) ).
% mult_cancel_right
thf(fact_260_div__0,axiom,
! [A2: real] :
( ( divide_divide_real @ zero_zero_real @ A2 )
= zero_zero_real ) ).
% div_0
thf(fact_261_div__0,axiom,
! [A2: nat] :
( ( divide_divide_nat @ zero_zero_nat @ A2 )
= zero_zero_nat ) ).
% div_0
thf(fact_262_div__0,axiom,
! [A2: int] :
( ( divide_divide_int @ zero_zero_int @ A2 )
= zero_zero_int ) ).
% div_0
thf(fact_263_div__by__0,axiom,
! [A2: real] :
( ( divide_divide_real @ A2 @ zero_zero_real )
= zero_zero_real ) ).
% div_by_0
thf(fact_264_div__by__0,axiom,
! [A2: nat] :
( ( divide_divide_nat @ A2 @ zero_zero_nat )
= zero_zero_nat ) ).
% div_by_0
thf(fact_265_div__by__0,axiom,
! [A2: int] :
( ( divide_divide_int @ A2 @ zero_zero_int )
= zero_zero_int ) ).
% div_by_0
thf(fact_266_of__nat__le__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% of_nat_le_iff
thf(fact_267_of__nat__le__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% of_nat_le_iff
thf(fact_268_of__nat__le__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% of_nat_le_iff
thf(fact_269_abs__0,axiom,
( ( abs_abs_real @ zero_zero_real )
= zero_zero_real ) ).
% abs_0
thf(fact_270_abs__0,axiom,
( ( abs_abs_int @ zero_zero_int )
= zero_zero_int ) ).
% abs_0
thf(fact_271_of__nat__mult,axiom,
! [M: nat,N: nat] :
( ( semiri1316708129612266289at_nat @ ( times_times_nat @ M @ N ) )
= ( times_times_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% of_nat_mult
thf(fact_272_of__nat__mult,axiom,
! [M: nat,N: nat] :
( ( semiri5074537144036343181t_real @ ( times_times_nat @ M @ N ) )
= ( times_times_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).
% of_nat_mult
thf(fact_273_of__nat__mult,axiom,
! [M: nat,N: nat] :
( ( semiri1314217659103216013at_int @ ( times_times_nat @ M @ N ) )
= ( times_times_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% of_nat_mult
thf(fact_274_abs__mult__self__eq,axiom,
! [A2: real] :
( ( times_times_real @ ( abs_abs_real @ A2 ) @ ( abs_abs_real @ A2 ) )
= ( times_times_real @ A2 @ A2 ) ) ).
% abs_mult_self_eq
thf(fact_275_abs__mult__self__eq,axiom,
! [A2: int] :
( ( times_times_int @ ( abs_abs_int @ A2 ) @ ( abs_abs_int @ A2 ) )
= ( times_times_int @ A2 @ A2 ) ) ).
% abs_mult_self_eq
thf(fact_276_abs__of__nat,axiom,
! [N: nat] :
( ( abs_abs_real @ ( semiri5074537144036343181t_real @ N ) )
= ( semiri5074537144036343181t_real @ N ) ) ).
% abs_of_nat
thf(fact_277_abs__of__nat,axiom,
! [N: nat] :
( ( abs_abs_int @ ( semiri1314217659103216013at_int @ N ) )
= ( semiri1314217659103216013at_int @ N ) ) ).
% abs_of_nat
thf(fact_278_sum_Oempty,axiom,
! [G: set_real > nat] :
( ( groups3012202523422989166al_nat @ G @ bot_bot_set_set_real )
= zero_zero_nat ) ).
% sum.empty
thf(fact_279_sum_Oempty,axiom,
! [G: set_real > int] :
( ( groups3009712052913938890al_int @ G @ bot_bot_set_set_real )
= zero_zero_int ) ).
% sum.empty
thf(fact_280_sum_Oempty,axiom,
! [G: real > real] :
( ( groups8097168146408367636l_real @ G @ bot_bot_set_real )
= zero_zero_real ) ).
% sum.empty
thf(fact_281_sum_Oempty,axiom,
! [G: real > nat] :
( ( groups1935376822645274424al_nat @ G @ bot_bot_set_real )
= zero_zero_nat ) ).
% sum.empty
thf(fact_282_sum_Oempty,axiom,
! [G: real > int] :
( ( groups1932886352136224148al_int @ G @ bot_bot_set_real )
= zero_zero_int ) ).
% sum.empty
thf(fact_283_sum_Oempty,axiom,
! [G: nat > real] :
( ( groups6591440286371151544t_real @ G @ bot_bot_set_nat )
= zero_zero_real ) ).
% sum.empty
thf(fact_284_sum_Oempty,axiom,
! [G: nat > nat] :
( ( groups3542108847815614940at_nat @ G @ bot_bot_set_nat )
= zero_zero_nat ) ).
% sum.empty
thf(fact_285_sum_Oempty,axiom,
! [G: nat > int] :
( ( groups3539618377306564664at_int @ G @ bot_bot_set_nat )
= zero_zero_int ) ).
% sum.empty
thf(fact_286_sum_Oempty,axiom,
! [G: set_real > real] :
( ( groups8702937949983641418l_real @ G @ bot_bot_set_set_real )
= zero_zero_real ) ).
% sum.empty
thf(fact_287_of__nat__0,axiom,
( ( semiri1316708129612266289at_nat @ zero_zero_nat )
= zero_zero_nat ) ).
% of_nat_0
thf(fact_288_of__nat__0,axiom,
( ( semiri5074537144036343181t_real @ zero_zero_nat )
= zero_zero_real ) ).
% of_nat_0
thf(fact_289_of__nat__0,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% of_nat_0
thf(fact_290_of__nat__0__eq__iff,axiom,
! [N: nat] :
( ( zero_zero_nat
= ( semiri1316708129612266289at_nat @ N ) )
= ( zero_zero_nat = N ) ) ).
% of_nat_0_eq_iff
thf(fact_291_of__nat__0__eq__iff,axiom,
! [N: nat] :
( ( zero_zero_real
= ( semiri5074537144036343181t_real @ N ) )
= ( zero_zero_nat = N ) ) ).
% of_nat_0_eq_iff
thf(fact_292_of__nat__0__eq__iff,axiom,
! [N: nat] :
( ( zero_zero_int
= ( semiri1314217659103216013at_int @ N ) )
= ( zero_zero_nat = N ) ) ).
% of_nat_0_eq_iff
thf(fact_293_of__nat__eq__0__iff,axiom,
! [M: nat] :
( ( ( semiri1316708129612266289at_nat @ M )
= zero_zero_nat )
= ( M = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_294_of__nat__eq__0__iff,axiom,
! [M: nat] :
( ( ( semiri5074537144036343181t_real @ M )
= zero_zero_real )
= ( M = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_295_of__nat__eq__0__iff,axiom,
! [M: nat] :
( ( ( semiri1314217659103216013at_int @ M )
= zero_zero_int )
= ( M = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_296_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_297_minus__nat_Odiff__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ zero_zero_nat )
= M ) ).
% minus_nat.diff_0
thf(fact_298_mult__0,axiom,
! [N: nat] :
( ( times_times_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% mult_0
thf(fact_299_bot__nat__0_Oextremum__unique,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
= ( A2 = zero_zero_nat ) ) ).
% bot_nat_0.extremum_unique
thf(fact_300_bot__nat__0_Oextremum__uniqueI,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
=> ( A2 = zero_zero_nat ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_301_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_302_diffs0__imp__equal,axiom,
! [M: nat,N: nat] :
( ( ( minus_minus_nat @ M @ N )
= zero_zero_nat )
=> ( ( ( minus_minus_nat @ N @ M )
= zero_zero_nat )
=> ( M = N ) ) ) ).
% diffs0_imp_equal
thf(fact_303_sum__subtractf__nat,axiom,
! [A: set_set_real,G: set_real > nat,F: set_real > nat] :
( ! [X3: set_real] :
( ( member_set_real @ X3 @ A )
=> ( ord_less_eq_nat @ ( G @ X3 ) @ ( F @ X3 ) ) )
=> ( ( groups3012202523422989166al_nat
@ ^ [X2: set_real] : ( minus_minus_nat @ ( F @ X2 ) @ ( G @ X2 ) )
@ A )
= ( minus_minus_nat @ ( groups3012202523422989166al_nat @ F @ A ) @ ( groups3012202523422989166al_nat @ G @ A ) ) ) ) ).
% sum_subtractf_nat
thf(fact_304_sum__subtractf__nat,axiom,
! [A: set_real,G: real > nat,F: real > nat] :
( ! [X3: real] :
( ( member_real @ X3 @ A )
=> ( ord_less_eq_nat @ ( G @ X3 ) @ ( F @ X3 ) ) )
=> ( ( groups1935376822645274424al_nat
@ ^ [X2: real] : ( minus_minus_nat @ ( F @ X2 ) @ ( G @ X2 ) )
@ A )
= ( minus_minus_nat @ ( groups1935376822645274424al_nat @ F @ A ) @ ( groups1935376822645274424al_nat @ G @ A ) ) ) ) ).
% sum_subtractf_nat
thf(fact_305_sum__subtractf__nat,axiom,
! [A: set_nat,G: nat > nat,F: nat > nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( ord_less_eq_nat @ ( G @ X3 ) @ ( F @ X3 ) ) )
=> ( ( groups3542108847815614940at_nat
@ ^ [X2: nat] : ( minus_minus_nat @ ( F @ X2 ) @ ( G @ X2 ) )
@ A )
= ( minus_minus_nat @ ( groups3542108847815614940at_nat @ F @ A ) @ ( groups3542108847815614940at_nat @ G @ A ) ) ) ) ).
% sum_subtractf_nat
thf(fact_306_sum__subtractf__nat,axiom,
! [A: set_o,G: $o > nat,F: $o > nat] :
( ! [X3: $o] :
( ( member_o @ X3 @ A )
=> ( ord_less_eq_nat @ ( G @ X3 ) @ ( F @ X3 ) ) )
=> ( ( groups8507830703676809646_o_nat
@ ^ [X2: $o] : ( minus_minus_nat @ ( F @ X2 ) @ ( G @ X2 ) )
@ A )
= ( minus_minus_nat @ ( groups8507830703676809646_o_nat @ F @ A ) @ ( groups8507830703676809646_o_nat @ G @ A ) ) ) ) ).
% sum_subtractf_nat
thf(fact_307_mult__not__zero,axiom,
! [A2: real,B: real] :
( ( ( times_times_real @ A2 @ B )
!= zero_zero_real )
=> ( ( A2 != zero_zero_real )
& ( B != zero_zero_real ) ) ) ).
% mult_not_zero
thf(fact_308_mult__not__zero,axiom,
! [A2: nat,B: nat] :
( ( ( times_times_nat @ A2 @ B )
!= zero_zero_nat )
=> ( ( A2 != zero_zero_nat )
& ( B != zero_zero_nat ) ) ) ).
% mult_not_zero
thf(fact_309_mult__not__zero,axiom,
! [A2: int,B: int] :
( ( ( times_times_int @ A2 @ B )
!= zero_zero_int )
=> ( ( A2 != zero_zero_int )
& ( B != zero_zero_int ) ) ) ).
% mult_not_zero
thf(fact_310_divisors__zero,axiom,
! [A2: real,B: real] :
( ( ( times_times_real @ A2 @ B )
= zero_zero_real )
=> ( ( A2 = zero_zero_real )
| ( B = zero_zero_real ) ) ) ).
% divisors_zero
thf(fact_311_divisors__zero,axiom,
! [A2: nat,B: nat] :
( ( ( times_times_nat @ A2 @ B )
= zero_zero_nat )
=> ( ( A2 = zero_zero_nat )
| ( B = zero_zero_nat ) ) ) ).
% divisors_zero
thf(fact_312_divisors__zero,axiom,
! [A2: int,B: int] :
( ( ( times_times_int @ A2 @ B )
= zero_zero_int )
=> ( ( A2 = zero_zero_int )
| ( B = zero_zero_int ) ) ) ).
% divisors_zero
thf(fact_313_no__zero__divisors,axiom,
! [A2: real,B: real] :
( ( A2 != zero_zero_real )
=> ( ( B != zero_zero_real )
=> ( ( times_times_real @ A2 @ B )
!= zero_zero_real ) ) ) ).
% no_zero_divisors
thf(fact_314_no__zero__divisors,axiom,
! [A2: nat,B: nat] :
( ( A2 != zero_zero_nat )
=> ( ( B != zero_zero_nat )
=> ( ( times_times_nat @ A2 @ B )
!= zero_zero_nat ) ) ) ).
% no_zero_divisors
thf(fact_315_no__zero__divisors,axiom,
! [A2: int,B: int] :
( ( A2 != zero_zero_int )
=> ( ( B != zero_zero_int )
=> ( ( times_times_int @ A2 @ B )
!= zero_zero_int ) ) ) ).
% no_zero_divisors
thf(fact_316_mult__left__cancel,axiom,
! [C: real,A2: real,B: real] :
( ( C != zero_zero_real )
=> ( ( ( times_times_real @ C @ A2 )
= ( times_times_real @ C @ B ) )
= ( A2 = B ) ) ) ).
% mult_left_cancel
thf(fact_317_mult__left__cancel,axiom,
! [C: nat,A2: nat,B: nat] :
( ( C != zero_zero_nat )
=> ( ( ( times_times_nat @ C @ A2 )
= ( times_times_nat @ C @ B ) )
= ( A2 = B ) ) ) ).
% mult_left_cancel
thf(fact_318_mult__left__cancel,axiom,
! [C: int,A2: int,B: int] :
( ( C != zero_zero_int )
=> ( ( ( times_times_int @ C @ A2 )
= ( times_times_int @ C @ B ) )
= ( A2 = B ) ) ) ).
% mult_left_cancel
thf(fact_319_mult__right__cancel,axiom,
! [C: real,A2: real,B: real] :
( ( C != zero_zero_real )
=> ( ( ( times_times_real @ A2 @ C )
= ( times_times_real @ B @ C ) )
= ( A2 = B ) ) ) ).
% mult_right_cancel
thf(fact_320_mult__right__cancel,axiom,
! [C: nat,A2: nat,B: nat] :
( ( C != zero_zero_nat )
=> ( ( ( times_times_nat @ A2 @ C )
= ( times_times_nat @ B @ C ) )
= ( A2 = B ) ) ) ).
% mult_right_cancel
thf(fact_321_mult__right__cancel,axiom,
! [C: int,A2: int,B: int] :
( ( C != zero_zero_int )
=> ( ( ( times_times_int @ A2 @ C )
= ( times_times_int @ B @ C ) )
= ( A2 = B ) ) ) ).
% mult_right_cancel
thf(fact_322_left__diff__distrib,axiom,
! [A2: real,B: real,C: real] :
( ( times_times_real @ ( minus_minus_real @ A2 @ B ) @ C )
= ( minus_minus_real @ ( times_times_real @ A2 @ C ) @ ( times_times_real @ B @ C ) ) ) ).
% left_diff_distrib
thf(fact_323_left__diff__distrib,axiom,
! [A2: int,B: int,C: int] :
( ( times_times_int @ ( minus_minus_int @ A2 @ B ) @ C )
= ( minus_minus_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B @ C ) ) ) ).
% left_diff_distrib
thf(fact_324_right__diff__distrib,axiom,
! [A2: real,B: real,C: real] :
( ( times_times_real @ A2 @ ( minus_minus_real @ B @ C ) )
= ( minus_minus_real @ ( times_times_real @ A2 @ B ) @ ( times_times_real @ A2 @ C ) ) ) ).
% right_diff_distrib
thf(fact_325_right__diff__distrib,axiom,
! [A2: int,B: int,C: int] :
( ( times_times_int @ A2 @ ( minus_minus_int @ B @ C ) )
= ( minus_minus_int @ ( times_times_int @ A2 @ B ) @ ( times_times_int @ A2 @ C ) ) ) ).
% right_diff_distrib
thf(fact_326_left__diff__distrib_H,axiom,
! [B: real,C: real,A2: real] :
( ( times_times_real @ ( minus_minus_real @ B @ C ) @ A2 )
= ( minus_minus_real @ ( times_times_real @ B @ A2 ) @ ( times_times_real @ C @ A2 ) ) ) ).
% left_diff_distrib'
thf(fact_327_left__diff__distrib_H,axiom,
! [B: nat,C: nat,A2: nat] :
( ( times_times_nat @ ( minus_minus_nat @ B @ C ) @ A2 )
= ( minus_minus_nat @ ( times_times_nat @ B @ A2 ) @ ( times_times_nat @ C @ A2 ) ) ) ).
% left_diff_distrib'
thf(fact_328_left__diff__distrib_H,axiom,
! [B: int,C: int,A2: int] :
( ( times_times_int @ ( minus_minus_int @ B @ C ) @ A2 )
= ( minus_minus_int @ ( times_times_int @ B @ A2 ) @ ( times_times_int @ C @ A2 ) ) ) ).
% left_diff_distrib'
thf(fact_329_right__diff__distrib_H,axiom,
! [A2: real,B: real,C: real] :
( ( times_times_real @ A2 @ ( minus_minus_real @ B @ C ) )
= ( minus_minus_real @ ( times_times_real @ A2 @ B ) @ ( times_times_real @ A2 @ C ) ) ) ).
% right_diff_distrib'
thf(fact_330_right__diff__distrib_H,axiom,
! [A2: nat,B: nat,C: nat] :
( ( times_times_nat @ A2 @ ( minus_minus_nat @ B @ C ) )
= ( minus_minus_nat @ ( times_times_nat @ A2 @ B ) @ ( times_times_nat @ A2 @ C ) ) ) ).
% right_diff_distrib'
thf(fact_331_right__diff__distrib_H,axiom,
! [A2: int,B: int,C: int] :
( ( times_times_int @ A2 @ ( minus_minus_int @ B @ C ) )
= ( minus_minus_int @ ( times_times_int @ A2 @ B ) @ ( times_times_int @ A2 @ C ) ) ) ).
% right_diff_distrib'
thf(fact_332_of__nat__mono,axiom,
! [I2: nat,J: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ I2 ) @ ( semiri5074537144036343181t_real @ J ) ) ) ).
% of_nat_mono
thf(fact_333_of__nat__mono,axiom,
! [I2: nat,J: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ I2 ) @ ( semiri1316708129612266289at_nat @ J ) ) ) ).
% of_nat_mono
thf(fact_334_of__nat__mono,axiom,
! [I2: nat,J: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ I2 ) @ ( semiri1314217659103216013at_int @ J ) ) ) ).
% of_nat_mono
thf(fact_335_abs__eq__0__iff,axiom,
! [A2: real] :
( ( ( abs_abs_real @ A2 )
= zero_zero_real )
= ( A2 = zero_zero_real ) ) ).
% abs_eq_0_iff
thf(fact_336_abs__eq__0__iff,axiom,
! [A2: int] :
( ( ( abs_abs_int @ A2 )
= zero_zero_int )
= ( A2 = zero_zero_int ) ) ).
% abs_eq_0_iff
thf(fact_337_mult__of__nat__commute,axiom,
! [X: nat,Y: nat] :
( ( times_times_nat @ ( semiri1316708129612266289at_nat @ X ) @ Y )
= ( times_times_nat @ Y @ ( semiri1316708129612266289at_nat @ X ) ) ) ).
% mult_of_nat_commute
thf(fact_338_mult__of__nat__commute,axiom,
! [X: nat,Y: real] :
( ( times_times_real @ ( semiri5074537144036343181t_real @ X ) @ Y )
= ( times_times_real @ Y @ ( semiri5074537144036343181t_real @ X ) ) ) ).
% mult_of_nat_commute
thf(fact_339_mult__of__nat__commute,axiom,
! [X: nat,Y: int] :
( ( times_times_int @ ( semiri1314217659103216013at_int @ X ) @ Y )
= ( times_times_int @ Y @ ( semiri1314217659103216013at_int @ X ) ) ) ).
% mult_of_nat_commute
thf(fact_340_of__nat__diff,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( ( semiri1316708129612266289at_nat @ ( minus_minus_nat @ M @ N ) )
= ( minus_minus_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ) ).
% of_nat_diff
thf(fact_341_of__nat__diff,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( ( semiri5074537144036343181t_real @ ( minus_minus_nat @ M @ N ) )
= ( minus_minus_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ).
% of_nat_diff
thf(fact_342_of__nat__diff,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ M @ N ) )
= ( minus_minus_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ) ).
% of_nat_diff
thf(fact_343_abs__mult,axiom,
! [A2: real,B: real] :
( ( abs_abs_real @ ( times_times_real @ A2 @ B ) )
= ( times_times_real @ ( abs_abs_real @ A2 ) @ ( abs_abs_real @ B ) ) ) ).
% abs_mult
thf(fact_344_abs__mult,axiom,
! [A2: int,B: int] :
( ( abs_abs_int @ ( times_times_int @ A2 @ B ) )
= ( times_times_int @ ( abs_abs_int @ A2 ) @ ( abs_abs_int @ B ) ) ) ).
% abs_mult
thf(fact_345_lambda__zero,axiom,
( ( ^ [H2: real] : zero_zero_real )
= ( times_times_real @ zero_zero_real ) ) ).
% lambda_zero
thf(fact_346_lambda__zero,axiom,
( ( ^ [H2: nat] : zero_zero_nat )
= ( times_times_nat @ zero_zero_nat ) ) ).
% lambda_zero
thf(fact_347_lambda__zero,axiom,
( ( ^ [H2: int] : zero_zero_int )
= ( times_times_int @ zero_zero_int ) ) ).
% lambda_zero
thf(fact_348_mult__mono,axiom,
! [A2: real,B: real,C: real,D: real] :
( ( ord_less_eq_real @ A2 @ B )
=> ( ( ord_less_eq_real @ C @ D )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ ( times_times_real @ A2 @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).
% mult_mono
thf(fact_349_mult__mono,axiom,
! [A2: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_mono
thf(fact_350_mult__mono,axiom,
! [A2: int,B: int,C: int,D: int] :
( ( ord_less_eq_int @ A2 @ B )
=> ( ( ord_less_eq_int @ C @ D )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% mult_mono
thf(fact_351_mult__mono_H,axiom,
! [A2: real,B: real,C: real,D: real] :
( ( ord_less_eq_real @ A2 @ B )
=> ( ( ord_less_eq_real @ C @ D )
=> ( ( ord_less_eq_real @ zero_zero_real @ A2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ ( times_times_real @ A2 @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).
% mult_mono'
thf(fact_352_mult__mono_H,axiom,
! [A2: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_mono'
thf(fact_353_mult__mono_H,axiom,
! [A2: int,B: int,C: int,D: int] :
( ( ord_less_eq_int @ A2 @ B )
=> ( ( ord_less_eq_int @ C @ D )
=> ( ( ord_less_eq_int @ zero_zero_int @ A2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% mult_mono'
thf(fact_354_zero__le__square,axiom,
! [A2: real] : ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A2 @ A2 ) ) ).
% zero_le_square
thf(fact_355_zero__le__square,axiom,
! [A2: int] : ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A2 @ A2 ) ) ).
% zero_le_square
thf(fact_356_split__mult__pos__le,axiom,
! [A2: real,B: real] :
( ( ( ( ord_less_eq_real @ zero_zero_real @ A2 )
& ( ord_less_eq_real @ zero_zero_real @ B ) )
| ( ( ord_less_eq_real @ A2 @ zero_zero_real )
& ( ord_less_eq_real @ B @ zero_zero_real ) ) )
=> ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A2 @ B ) ) ) ).
% split_mult_pos_le
thf(fact_357_split__mult__pos__le,axiom,
! [A2: int,B: int] :
( ( ( ( ord_less_eq_int @ zero_zero_int @ A2 )
& ( ord_less_eq_int @ zero_zero_int @ B ) )
| ( ( ord_less_eq_int @ A2 @ zero_zero_int )
& ( ord_less_eq_int @ B @ zero_zero_int ) ) )
=> ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A2 @ B ) ) ) ).
% split_mult_pos_le
thf(fact_358_mult__left__mono__neg,axiom,
! [B: real,A2: real,C: real] :
( ( ord_less_eq_real @ B @ A2 )
=> ( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ C @ A2 ) @ ( times_times_real @ C @ B ) ) ) ) ).
% mult_left_mono_neg
thf(fact_359_mult__left__mono__neg,axiom,
! [B: int,A2: int,C: int] :
( ( ord_less_eq_int @ B @ A2 )
=> ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B ) ) ) ) ).
% mult_left_mono_neg
thf(fact_360_mult__nonpos__nonpos,axiom,
! [A2: real,B: real] :
( ( ord_less_eq_real @ A2 @ zero_zero_real )
=> ( ( ord_less_eq_real @ B @ zero_zero_real )
=> ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A2 @ B ) ) ) ) ).
% mult_nonpos_nonpos
thf(fact_361_mult__nonpos__nonpos,axiom,
! [A2: int,B: int] :
( ( ord_less_eq_int @ A2 @ zero_zero_int )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A2 @ B ) ) ) ) ).
% mult_nonpos_nonpos
thf(fact_362_mult__left__mono,axiom,
! [A2: real,B: real,C: real] :
( ( ord_less_eq_real @ A2 @ B )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ ( times_times_real @ C @ A2 ) @ ( times_times_real @ C @ B ) ) ) ) ).
% mult_left_mono
thf(fact_363_mult__left__mono,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ C @ A2 ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% mult_left_mono
thf(fact_364_mult__left__mono,axiom,
! [A2: int,B: int,C: int] :
( ( ord_less_eq_int @ A2 @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B ) ) ) ) ).
% mult_left_mono
thf(fact_365_mult__right__mono__neg,axiom,
! [B: real,A2: real,C: real] :
( ( ord_less_eq_real @ B @ A2 )
=> ( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ A2 @ C ) @ ( times_times_real @ B @ C ) ) ) ) ).
% mult_right_mono_neg
thf(fact_366_mult__right__mono__neg,axiom,
! [B: int,A2: int,C: int] :
( ( ord_less_eq_int @ B @ A2 )
=> ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% mult_right_mono_neg
thf(fact_367_mult__right__mono,axiom,
! [A2: real,B: real,C: real] :
( ( ord_less_eq_real @ A2 @ B )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ ( times_times_real @ A2 @ C ) @ ( times_times_real @ B @ C ) ) ) ) ).
% mult_right_mono
thf(fact_368_mult__right__mono,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% mult_right_mono
thf(fact_369_mult__right__mono,axiom,
! [A2: int,B: int,C: int] :
( ( ord_less_eq_int @ A2 @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% mult_right_mono
thf(fact_370_mult__le__0__iff,axiom,
! [A2: real,B: real] :
( ( ord_less_eq_real @ ( times_times_real @ A2 @ B ) @ zero_zero_real )
= ( ( ( ord_less_eq_real @ zero_zero_real @ A2 )
& ( ord_less_eq_real @ B @ zero_zero_real ) )
| ( ( ord_less_eq_real @ A2 @ zero_zero_real )
& ( ord_less_eq_real @ zero_zero_real @ B ) ) ) ) ).
% mult_le_0_iff
thf(fact_371_mult__le__0__iff,axiom,
! [A2: int,B: int] :
( ( ord_less_eq_int @ ( times_times_int @ A2 @ B ) @ zero_zero_int )
= ( ( ( ord_less_eq_int @ zero_zero_int @ A2 )
& ( ord_less_eq_int @ B @ zero_zero_int ) )
| ( ( ord_less_eq_int @ A2 @ zero_zero_int )
& ( ord_less_eq_int @ zero_zero_int @ B ) ) ) ) ).
% mult_le_0_iff
thf(fact_372_split__mult__neg__le,axiom,
! [A2: real,B: real] :
( ( ( ( ord_less_eq_real @ zero_zero_real @ A2 )
& ( ord_less_eq_real @ B @ zero_zero_real ) )
| ( ( ord_less_eq_real @ A2 @ zero_zero_real )
& ( ord_less_eq_real @ zero_zero_real @ B ) ) )
=> ( ord_less_eq_real @ ( times_times_real @ A2 @ B ) @ zero_zero_real ) ) ).
% split_mult_neg_le
thf(fact_373_split__mult__neg__le,axiom,
! [A2: nat,B: nat] :
( ( ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
& ( ord_less_eq_nat @ B @ zero_zero_nat ) )
| ( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
& ( ord_less_eq_nat @ zero_zero_nat @ B ) ) )
=> ( ord_less_eq_nat @ ( times_times_nat @ A2 @ B ) @ zero_zero_nat ) ) ).
% split_mult_neg_le
thf(fact_374_split__mult__neg__le,axiom,
! [A2: int,B: int] :
( ( ( ( ord_less_eq_int @ zero_zero_int @ A2 )
& ( ord_less_eq_int @ B @ zero_zero_int ) )
| ( ( ord_less_eq_int @ A2 @ zero_zero_int )
& ( ord_less_eq_int @ zero_zero_int @ B ) ) )
=> ( ord_less_eq_int @ ( times_times_int @ A2 @ B ) @ zero_zero_int ) ) ).
% split_mult_neg_le
thf(fact_375_mult__nonneg__nonneg,axiom,
! [A2: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ A2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A2 @ B ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_376_mult__nonneg__nonneg,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( times_times_nat @ A2 @ B ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_377_mult__nonneg__nonneg,axiom,
! [A2: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A2 @ B ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_378_mult__nonneg__nonpos,axiom,
! [A2: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ A2 )
=> ( ( ord_less_eq_real @ B @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ A2 @ B ) @ zero_zero_real ) ) ) ).
% mult_nonneg_nonpos
thf(fact_379_mult__nonneg__nonpos,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ A2 @ B ) @ zero_zero_nat ) ) ) ).
% mult_nonneg_nonpos
thf(fact_380_mult__nonneg__nonpos,axiom,
! [A2: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A2 )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ A2 @ B ) @ zero_zero_int ) ) ) ).
% mult_nonneg_nonpos
thf(fact_381_mult__nonpos__nonneg,axiom,
! [A2: real,B: real] :
( ( ord_less_eq_real @ A2 @ zero_zero_real )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ord_less_eq_real @ ( times_times_real @ A2 @ B ) @ zero_zero_real ) ) ) ).
% mult_nonpos_nonneg
thf(fact_382_mult__nonpos__nonneg,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ ( times_times_nat @ A2 @ B ) @ zero_zero_nat ) ) ) ).
% mult_nonpos_nonneg
thf(fact_383_mult__nonpos__nonneg,axiom,
! [A2: int,B: int] :
( ( ord_less_eq_int @ A2 @ zero_zero_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ ( times_times_int @ A2 @ B ) @ zero_zero_int ) ) ) ).
% mult_nonpos_nonneg
thf(fact_384_mult__nonneg__nonpos2,axiom,
! [A2: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ A2 )
=> ( ( ord_less_eq_real @ B @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ B @ A2 ) @ zero_zero_real ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_385_mult__nonneg__nonpos2,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ B @ A2 ) @ zero_zero_nat ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_386_mult__nonneg__nonpos2,axiom,
! [A2: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A2 )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ B @ A2 ) @ zero_zero_int ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_387_zero__le__mult__iff,axiom,
! [A2: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A2 @ B ) )
= ( ( ( ord_less_eq_real @ zero_zero_real @ A2 )
& ( ord_less_eq_real @ zero_zero_real @ B ) )
| ( ( ord_less_eq_real @ A2 @ zero_zero_real )
& ( ord_less_eq_real @ B @ zero_zero_real ) ) ) ) ).
% zero_le_mult_iff
thf(fact_388_zero__le__mult__iff,axiom,
! [A2: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A2 @ B ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ A2 )
& ( ord_less_eq_int @ zero_zero_int @ B ) )
| ( ( ord_less_eq_int @ A2 @ zero_zero_int )
& ( ord_less_eq_int @ B @ zero_zero_int ) ) ) ) ).
% zero_le_mult_iff
thf(fact_389_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A2: real,B: real,C: real] :
( ( ord_less_eq_real @ A2 @ B )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ ( times_times_real @ C @ A2 ) @ ( times_times_real @ C @ B ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_390_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ C @ A2 ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_391_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A2: int,B: int,C: int] :
( ( ord_less_eq_int @ A2 @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_392_of__nat__0__le__iff,axiom,
! [N: nat] : ( ord_less_eq_real @ zero_zero_real @ ( semiri5074537144036343181t_real @ N ) ) ).
% of_nat_0_le_iff
thf(fact_393_of__nat__0__le__iff,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N ) ) ).
% of_nat_0_le_iff
thf(fact_394_of__nat__0__le__iff,axiom,
! [N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N ) ) ).
% of_nat_0_le_iff
thf(fact_395_abs__mult__pos,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( times_times_real @ ( abs_abs_real @ Y ) @ X )
= ( abs_abs_real @ ( times_times_real @ Y @ X ) ) ) ) ).
% abs_mult_pos
thf(fact_396_abs__mult__pos,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( times_times_int @ ( abs_abs_int @ Y ) @ X )
= ( abs_abs_int @ ( times_times_int @ Y @ X ) ) ) ) ).
% abs_mult_pos
thf(fact_397_abs__eq__mult,axiom,
! [A2: real,B: real] :
( ( ( ( ord_less_eq_real @ zero_zero_real @ A2 )
| ( ord_less_eq_real @ A2 @ zero_zero_real ) )
& ( ( ord_less_eq_real @ zero_zero_real @ B )
| ( ord_less_eq_real @ B @ zero_zero_real ) ) )
=> ( ( abs_abs_real @ ( times_times_real @ A2 @ B ) )
= ( times_times_real @ ( abs_abs_real @ A2 ) @ ( abs_abs_real @ B ) ) ) ) ).
% abs_eq_mult
thf(fact_398_abs__eq__mult,axiom,
! [A2: int,B: int] :
( ( ( ( ord_less_eq_int @ zero_zero_int @ A2 )
| ( ord_less_eq_int @ A2 @ zero_zero_int ) )
& ( ( ord_less_eq_int @ zero_zero_int @ B )
| ( ord_less_eq_int @ B @ zero_zero_int ) ) )
=> ( ( abs_abs_int @ ( times_times_int @ A2 @ B ) )
= ( times_times_int @ ( abs_abs_int @ A2 ) @ ( abs_abs_int @ B ) ) ) ) ).
% abs_eq_mult
thf(fact_399_abs__mult__pos_H,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( times_times_real @ X @ ( abs_abs_real @ Y ) )
= ( abs_abs_real @ ( times_times_real @ X @ Y ) ) ) ) ).
% abs_mult_pos'
thf(fact_400_abs__mult__pos_H,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( times_times_int @ X @ ( abs_abs_int @ Y ) )
= ( abs_abs_int @ ( times_times_int @ X @ Y ) ) ) ) ).
% abs_mult_pos'
thf(fact_401_f12,axiom,
( hensto240673015341029504l_real
@ ^ [X2: real] : ( minus_minus_real @ ( f2 @ X2 ) @ ( f1 @ X2 ) )
@ ( groups8702937949983641418l_real
@ ^ [K3: set_real] : ( times_times_real @ ( minus_minus_real @ ( f @ ( comple1385675409528146559p_real @ K3 ) ) @ ( f @ ( comple4887499456419720421f_real @ K3 ) ) ) @ ( divide_divide_real @ a @ ( semiri5074537144036343181t_real @ n ) ) )
@ ( regular_division @ zero_zero_real @ a @ n ) )
@ ( set_or1222579329274155063t_real @ zero_zero_real @ a ) ) ).
% f12
thf(fact_402_has__integral__empty,axiom,
! [F: real > real] : ( hensto240673015341029504l_real @ F @ zero_zero_real @ bot_bot_set_real ) ).
% has_integral_empty
thf(fact_403_has__integral__empty__eq,axiom,
! [F: real > real,I2: real] :
( ( hensto240673015341029504l_real @ F @ I2 @ bot_bot_set_real )
= ( I2 = zero_zero_real ) ) ).
% has_integral_empty_eq
thf(fact_404_ccSup__empty,axiom,
( ( comple5917660045593844715t_real @ bot_bo3378928929837779682t_real )
= bot_bot_set_set_real ) ).
% ccSup_empty
thf(fact_405_ccSup__empty,axiom,
( ( comple7399068483239264473et_nat @ bot_bot_set_set_nat )
= bot_bot_set_nat ) ).
% ccSup_empty
thf(fact_406_ccSup__empty,axiom,
( ( comple3096694443085538997t_real @ bot_bot_set_set_real )
= bot_bot_set_real ) ).
% ccSup_empty
thf(fact_407_ccSup__empty,axiom,
( ( complete_Sup_Sup_o @ bot_bot_set_o )
= bot_bot_o ) ).
% ccSup_empty
thf(fact_408_Sup__empty,axiom,
( ( comple5917660045593844715t_real @ bot_bo3378928929837779682t_real )
= bot_bot_set_set_real ) ).
% Sup_empty
thf(fact_409_Sup__empty,axiom,
( ( comple7399068483239264473et_nat @ bot_bot_set_set_nat )
= bot_bot_set_nat ) ).
% Sup_empty
thf(fact_410_Sup__empty,axiom,
( ( comple3096694443085538997t_real @ bot_bot_set_set_real )
= bot_bot_set_real ) ).
% Sup_empty
thf(fact_411_Sup__empty,axiom,
( ( complete_Sup_Sup_o @ bot_bot_set_o )
= bot_bot_o ) ).
% Sup_empty
thf(fact_412_sum__clauses_I1_J,axiom,
! [F: set_real > nat] :
( ( groups3012202523422989166al_nat @ F @ bot_bot_set_set_real )
= zero_zero_nat ) ).
% sum_clauses(1)
thf(fact_413_sum__clauses_I1_J,axiom,
! [F: set_real > int] :
( ( groups3009712052913938890al_int @ F @ bot_bot_set_set_real )
= zero_zero_int ) ).
% sum_clauses(1)
thf(fact_414_sum__clauses_I1_J,axiom,
! [F: real > real] :
( ( groups8097168146408367636l_real @ F @ bot_bot_set_real )
= zero_zero_real ) ).
% sum_clauses(1)
thf(fact_415_sum__clauses_I1_J,axiom,
! [F: real > nat] :
( ( groups1935376822645274424al_nat @ F @ bot_bot_set_real )
= zero_zero_nat ) ).
% sum_clauses(1)
thf(fact_416_sum__clauses_I1_J,axiom,
! [F: real > int] :
( ( groups1932886352136224148al_int @ F @ bot_bot_set_real )
= zero_zero_int ) ).
% sum_clauses(1)
thf(fact_417_sum__clauses_I1_J,axiom,
! [F: nat > real] :
( ( groups6591440286371151544t_real @ F @ bot_bot_set_nat )
= zero_zero_real ) ).
% sum_clauses(1)
thf(fact_418_sum__clauses_I1_J,axiom,
! [F: nat > nat] :
( ( groups3542108847815614940at_nat @ F @ bot_bot_set_nat )
= zero_zero_nat ) ).
% sum_clauses(1)
thf(fact_419_sum__clauses_I1_J,axiom,
! [F: nat > int] :
( ( groups3539618377306564664at_int @ F @ bot_bot_set_nat )
= zero_zero_int ) ).
% sum_clauses(1)
thf(fact_420_sum__clauses_I1_J,axiom,
! [F: set_real > real] :
( ( groups8702937949983641418l_real @ F @ bot_bot_set_set_real )
= zero_zero_real ) ).
% sum_clauses(1)
thf(fact_421_has__integral__0__eq,axiom,
! [I2: real,S: set_real] :
( ( hensto240673015341029504l_real
@ ^ [X2: real] : zero_zero_real
@ I2
@ S )
= ( I2 = zero_zero_real ) ) ).
% has_integral_0_eq
thf(fact_422_has__integral__restrict,axiom,
! [S: set_real,T: set_real,F: real > real,I2: real] :
( ( ord_less_eq_set_real @ S @ T )
=> ( ( hensto240673015341029504l_real
@ ^ [X2: real] : ( if_real @ ( member_real @ X2 @ S ) @ ( F @ X2 ) @ zero_zero_real )
@ I2
@ T )
= ( hensto240673015341029504l_real @ F @ I2 @ S ) ) ) ).
% has_integral_restrict
thf(fact_423_real__divide__square__eq,axiom,
! [R: real,A2: real] :
( ( divide_divide_real @ ( times_times_real @ R @ A2 ) @ ( times_times_real @ R @ R ) )
= ( divide_divide_real @ A2 @ R ) ) ).
% real_divide_square_eq
thf(fact_424_Sup__bot__conv_I1_J,axiom,
! [A: set_set_set_real] :
( ( ( comple5917660045593844715t_real @ A )
= bot_bot_set_set_real )
= ( ! [X2: set_set_real] :
( ( member_set_set_real @ X2 @ A )
=> ( X2 = bot_bot_set_set_real ) ) ) ) ).
% Sup_bot_conv(1)
thf(fact_425_Sup__bot__conv_I1_J,axiom,
! [A: set_set_nat] :
( ( ( comple7399068483239264473et_nat @ A )
= bot_bot_set_nat )
= ( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ A )
=> ( X2 = bot_bot_set_nat ) ) ) ) ).
% Sup_bot_conv(1)
thf(fact_426_Sup__bot__conv_I1_J,axiom,
! [A: set_set_real] :
( ( ( comple3096694443085538997t_real @ A )
= bot_bot_set_real )
= ( ! [X2: set_real] :
( ( member_set_real @ X2 @ A )
=> ( X2 = bot_bot_set_real ) ) ) ) ).
% Sup_bot_conv(1)
thf(fact_427_Sup__bot__conv_I1_J,axiom,
! [A: set_o] :
( ( ( complete_Sup_Sup_o @ A )
= bot_bot_o )
= ( ! [X2: $o] :
( ( member_o @ X2 @ A )
=> ( X2 = bot_bot_o ) ) ) ) ).
% Sup_bot_conv(1)
thf(fact_428_Sup__bot__conv_I2_J,axiom,
! [A: set_set_set_real] :
( ( bot_bot_set_set_real
= ( comple5917660045593844715t_real @ A ) )
= ( ! [X2: set_set_real] :
( ( member_set_set_real @ X2 @ A )
=> ( X2 = bot_bot_set_set_real ) ) ) ) ).
% Sup_bot_conv(2)
thf(fact_429_Sup__bot__conv_I2_J,axiom,
! [A: set_set_nat] :
( ( bot_bot_set_nat
= ( comple7399068483239264473et_nat @ A ) )
= ( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ A )
=> ( X2 = bot_bot_set_nat ) ) ) ) ).
% Sup_bot_conv(2)
thf(fact_430_Sup__bot__conv_I2_J,axiom,
! [A: set_set_real] :
( ( bot_bot_set_real
= ( comple3096694443085538997t_real @ A ) )
= ( ! [X2: set_real] :
( ( member_set_real @ X2 @ A )
=> ( X2 = bot_bot_set_real ) ) ) ) ).
% Sup_bot_conv(2)
thf(fact_431_Sup__bot__conv_I2_J,axiom,
! [A: set_o] :
( ( bot_bot_o
= ( complete_Sup_Sup_o @ A ) )
= ( ! [X2: $o] :
( ( member_o @ X2 @ A )
=> ( X2 = bot_bot_o ) ) ) ) ).
% Sup_bot_conv(2)
thf(fact_432_diff__diff__cancel,axiom,
! [I2: nat,N: nat] :
( ( ord_less_eq_nat @ I2 @ N )
=> ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I2 ) )
= I2 ) ) ).
% diff_diff_cancel
thf(fact_433_zle__int,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% zle_int
thf(fact_434_le__cube,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ ( times_times_nat @ M @ M ) ) ) ).
% le_cube
thf(fact_435_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_436_le__trans,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( ord_less_eq_nat @ J @ K )
=> ( ord_less_eq_nat @ I2 @ K ) ) ) ).
% le_trans
thf(fact_437_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_438_le__square,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ M ) ) ).
% le_square
thf(fact_439_le__antisym,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( M = N ) ) ) ).
% le_antisym
thf(fact_440_eq__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ( minus_minus_nat @ M @ K )
= ( minus_minus_nat @ N @ K ) )
= ( M = N ) ) ) ) ).
% eq_diff_iff
thf(fact_441_le__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% le_diff_iff
thf(fact_442_Nat_Odiff__diff__eq,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( minus_minus_nat @ M @ N ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_443_diff__le__mono,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M @ L ) @ ( minus_minus_nat @ N @ L ) ) ) ).
% diff_le_mono
thf(fact_444_diff__le__self,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).
% diff_le_self
thf(fact_445_le__diff__iff_H,axiom,
! [A2: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ C )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A2 ) @ ( minus_minus_nat @ C @ B ) )
= ( ord_less_eq_nat @ B @ A2 ) ) ) ) ).
% le_diff_iff'
thf(fact_446_mult__le__mono,axiom,
! [I2: nat,J: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( times_times_nat @ I2 @ K ) @ ( times_times_nat @ J @ L ) ) ) ) ).
% mult_le_mono
thf(fact_447_diff__le__mono2,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ).
% diff_le_mono2
thf(fact_448_mult__le__mono1,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ I2 @ K ) @ ( times_times_nat @ J @ K ) ) ) ).
% mult_le_mono1
thf(fact_449_mult__le__mono2,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ K @ I2 ) @ ( times_times_nat @ K @ J ) ) ) ).
% mult_le_mono2
thf(fact_450_nat__le__linear,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
| ( ord_less_eq_nat @ N @ M ) ) ).
% nat_le_linear
thf(fact_451_nonneg__int__cases,axiom,
! [K: int] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ~ ! [N3: nat] :
( K
!= ( semiri1314217659103216013at_int @ N3 ) ) ) ).
% nonneg_int_cases
thf(fact_452_diff__mult__distrib,axiom,
! [M: nat,N: nat,K: nat] :
( ( times_times_nat @ ( minus_minus_nat @ M @ N ) @ K )
= ( minus_minus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) ) ) ).
% diff_mult_distrib
thf(fact_453_zero__le__imp__eq__int,axiom,
! [K: int] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ? [N3: nat] :
( K
= ( semiri1314217659103216013at_int @ N3 ) ) ) ).
% zero_le_imp_eq_int
thf(fact_454_diff__mult__distrib2,axiom,
! [K: nat,M: nat,N: nat] :
( ( times_times_nat @ K @ ( minus_minus_nat @ M @ N ) )
= ( minus_minus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) ) ) ).
% diff_mult_distrib2
thf(fact_455_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B: nat] :
( ( P @ K )
=> ( ! [Y2: nat] :
( ( P @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ B ) )
=> ? [X3: nat] :
( ( P @ X3 )
& ! [Y4: nat] :
( ( P @ Y4 )
=> ( ord_less_eq_nat @ Y4 @ X3 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_456_zdiv__int,axiom,
! [M: nat,N: nat] :
( ( semiri1314217659103216013at_int @ ( divide_divide_nat @ M @ N ) )
= ( divide_divide_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% zdiv_int
thf(fact_457_div__le__mono,axiom,
! [M: nat,N: nat,K: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( divide_divide_nat @ M @ K ) @ ( divide_divide_nat @ N @ K ) ) ) ).
% div_le_mono
thf(fact_458_div__mult2__eq,axiom,
! [M: nat,N: nat,Q: nat] :
( ( divide_divide_nat @ M @ ( times_times_nat @ N @ Q ) )
= ( divide_divide_nat @ ( divide_divide_nat @ M @ N ) @ Q ) ) ).
% div_mult2_eq
thf(fact_459_div__le__dividend,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( divide_divide_nat @ M @ N ) @ M ) ).
% div_le_dividend
thf(fact_460_div__times__less__eq__dividend,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( times_times_nat @ ( divide_divide_nat @ M @ N ) @ N ) @ M ) ).
% div_times_less_eq_dividend
thf(fact_461_times__div__less__eq__dividend,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ ( times_times_nat @ N @ ( divide_divide_nat @ M @ N ) ) @ M ) ).
% times_div_less_eq_dividend
thf(fact_462_empty__Union__conv,axiom,
! [A: set_set_set_real] :
( ( bot_bot_set_set_real
= ( comple5917660045593844715t_real @ A ) )
= ( ! [X2: set_set_real] :
( ( member_set_set_real @ X2 @ A )
=> ( X2 = bot_bot_set_set_real ) ) ) ) ).
% empty_Union_conv
thf(fact_463_empty__Union__conv,axiom,
! [A: set_set_nat] :
( ( bot_bot_set_nat
= ( comple7399068483239264473et_nat @ A ) )
= ( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ A )
=> ( X2 = bot_bot_set_nat ) ) ) ) ).
% empty_Union_conv
thf(fact_464_empty__Union__conv,axiom,
! [A: set_set_real] :
( ( bot_bot_set_real
= ( comple3096694443085538997t_real @ A ) )
= ( ! [X2: set_real] :
( ( member_set_real @ X2 @ A )
=> ( X2 = bot_bot_set_real ) ) ) ) ).
% empty_Union_conv
thf(fact_465_Union__empty__conv,axiom,
! [A: set_set_set_real] :
( ( ( comple5917660045593844715t_real @ A )
= bot_bot_set_set_real )
= ( ! [X2: set_set_real] :
( ( member_set_set_real @ X2 @ A )
=> ( X2 = bot_bot_set_set_real ) ) ) ) ).
% Union_empty_conv
thf(fact_466_Union__empty__conv,axiom,
! [A: set_set_nat] :
( ( ( comple7399068483239264473et_nat @ A )
= bot_bot_set_nat )
= ( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ A )
=> ( X2 = bot_bot_set_nat ) ) ) ) ).
% Union_empty_conv
thf(fact_467_Union__empty__conv,axiom,
! [A: set_set_real] :
( ( ( comple3096694443085538997t_real @ A )
= bot_bot_set_real )
= ( ! [X2: set_real] :
( ( member_set_real @ X2 @ A )
=> ( X2 = bot_bot_set_real ) ) ) ) ).
% Union_empty_conv
thf(fact_468_int__diff__cases,axiom,
! [Z2: int] :
~ ! [M2: nat,N3: nat] :
( Z2
!= ( minus_minus_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N3 ) ) ) ).
% int_diff_cases
thf(fact_469_diff__commute,axiom,
! [I2: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I2 @ J ) @ K )
= ( minus_minus_nat @ ( minus_minus_nat @ I2 @ K ) @ J ) ) ).
% diff_commute
thf(fact_470_int__int__eq,axiom,
! [M: nat,N: nat] :
( ( ( semiri1314217659103216013at_int @ M )
= ( semiri1314217659103216013at_int @ N ) )
= ( M = N ) ) ).
% int_int_eq
thf(fact_471_Union__subsetI,axiom,
! [A: set_set_real,B4: set_set_real] :
( ! [X3: set_real] :
( ( member_set_real @ X3 @ A )
=> ? [Y4: set_real] :
( ( member_set_real @ Y4 @ B4 )
& ( ord_less_eq_set_real @ X3 @ Y4 ) ) )
=> ( ord_less_eq_set_real @ ( comple3096694443085538997t_real @ A ) @ ( comple3096694443085538997t_real @ B4 ) ) ) ).
% Union_subsetI
thf(fact_472_Union__upper,axiom,
! [B4: set_real,A: set_set_real] :
( ( member_set_real @ B4 @ A )
=> ( ord_less_eq_set_real @ B4 @ ( comple3096694443085538997t_real @ A ) ) ) ).
% Union_upper
thf(fact_473_Union__least,axiom,
! [A: set_set_real,C2: set_real] :
( ! [X5: set_real] :
( ( member_set_real @ X5 @ A )
=> ( ord_less_eq_set_real @ X5 @ C2 ) )
=> ( ord_less_eq_set_real @ ( comple3096694443085538997t_real @ A ) @ C2 ) ) ).
% Union_least
thf(fact_474_Union__mono,axiom,
! [A: set_set_real,B4: set_set_real] :
( ( ord_le3558479182127378552t_real @ A @ B4 )
=> ( ord_less_eq_set_real @ ( comple3096694443085538997t_real @ A ) @ ( comple3096694443085538997t_real @ B4 ) ) ) ).
% Union_mono
thf(fact_475_Union__regular__division,axiom,
! [A2: real,B: real,N: nat] :
( ( ord_less_eq_real @ A2 @ B )
=> ( ( ( N = zero_zero_nat )
=> ( ( comple3096694443085538997t_real @ ( regular_division @ A2 @ B @ N ) )
= bot_bot_set_real ) )
& ( ( N != zero_zero_nat )
=> ( ( comple3096694443085538997t_real @ ( regular_division @ A2 @ B @ N ) )
= ( set_or1222579329274155063t_real @ A2 @ B ) ) ) ) ) ).
% Union_regular_division
thf(fact_476_bot__nat__def,axiom,
bot_bot_nat = zero_zero_nat ).
% bot_nat_def
thf(fact_477_Union__empty,axiom,
( ( comple5917660045593844715t_real @ bot_bo3378928929837779682t_real )
= bot_bot_set_set_real ) ).
% Union_empty
thf(fact_478_Union__empty,axiom,
( ( comple7399068483239264473et_nat @ bot_bot_set_set_nat )
= bot_bot_set_nat ) ).
% Union_empty
thf(fact_479_Union__empty,axiom,
( ( comple3096694443085538997t_real @ bot_bot_set_set_real )
= bot_bot_set_real ) ).
% Union_empty
thf(fact_480_interval__bounds__real_I1_J,axiom,
! [A2: real,B: real] :
( ( ord_less_eq_real @ A2 @ B )
=> ( ( comple1385675409528146559p_real @ ( set_or1222579329274155063t_real @ A2 @ B ) )
= B ) ) ).
% interval_bounds_real(1)
thf(fact_481_interval__bounds__real_I2_J,axiom,
! [A2: real,B: real] :
( ( ord_less_eq_real @ A2 @ B )
=> ( ( comple4887499456419720421f_real @ ( set_or1222579329274155063t_real @ A2 @ B ) )
= A2 ) ) ).
% interval_bounds_real(2)
thf(fact_482_has__integral__eq,axiom,
! [S2: set_real,F: real > real,G: real > real,K: real] :
( ! [X3: real] :
( ( member_real @ X3 @ S2 )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( hensto240673015341029504l_real @ F @ K @ S2 )
=> ( hensto240673015341029504l_real @ G @ K @ S2 ) ) ) ).
% has_integral_eq
thf(fact_483_has__integral__cong,axiom,
! [S2: set_real,F: real > real,G: real > real,I2: real] :
( ! [X3: real] :
( ( member_real @ X3 @ S2 )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( hensto240673015341029504l_real @ F @ I2 @ S2 )
= ( hensto240673015341029504l_real @ G @ I2 @ S2 ) ) ) ).
% has_integral_cong
thf(fact_484_has__integral__eq__rhs,axiom,
! [F: real > real,J: real,S: set_real,I2: real] :
( ( hensto240673015341029504l_real @ F @ J @ S )
=> ( ( I2 = J )
=> ( hensto240673015341029504l_real @ F @ I2 @ S ) ) ) ).
% has_integral_eq_rhs
thf(fact_485_has__integral__unique,axiom,
! [F: real > real,K1: real,I2: set_real,K22: real] :
( ( hensto240673015341029504l_real @ F @ K1 @ I2 )
=> ( ( hensto240673015341029504l_real @ F @ K22 @ I2 )
=> ( K1 = K22 ) ) ) ).
% has_integral_unique
thf(fact_486_Sup__subset__mono,axiom,
! [A: set_set_real,B4: set_set_real] :
( ( ord_le3558479182127378552t_real @ A @ B4 )
=> ( ord_less_eq_set_real @ ( comple3096694443085538997t_real @ A ) @ ( comple3096694443085538997t_real @ B4 ) ) ) ).
% Sup_subset_mono
thf(fact_487_Sup__subset__mono,axiom,
! [A: set_o,B4: set_o] :
( ( ord_less_eq_set_o @ A @ B4 )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ A ) @ ( complete_Sup_Sup_o @ B4 ) ) ) ).
% Sup_subset_mono
thf(fact_488_Sup__upper2,axiom,
! [U2: set_real,A: set_set_real,V: set_real] :
( ( member_set_real @ U2 @ A )
=> ( ( ord_less_eq_set_real @ V @ U2 )
=> ( ord_less_eq_set_real @ V @ ( comple3096694443085538997t_real @ A ) ) ) ) ).
% Sup_upper2
thf(fact_489_Sup__upper2,axiom,
! [U2: $o,A: set_o,V: $o] :
( ( member_o @ U2 @ A )
=> ( ( ord_less_eq_o @ V @ U2 )
=> ( ord_less_eq_o @ V @ ( complete_Sup_Sup_o @ A ) ) ) ) ).
% Sup_upper2
thf(fact_490_Sup__le__iff,axiom,
! [A: set_set_real,B: set_real] :
( ( ord_less_eq_set_real @ ( comple3096694443085538997t_real @ A ) @ B )
= ( ! [X2: set_real] :
( ( member_set_real @ X2 @ A )
=> ( ord_less_eq_set_real @ X2 @ B ) ) ) ) ).
% Sup_le_iff
thf(fact_491_Sup__le__iff,axiom,
! [A: set_o,B: $o] :
( ( ord_less_eq_o @ ( complete_Sup_Sup_o @ A ) @ B )
= ( ! [X2: $o] :
( ( member_o @ X2 @ A )
=> ( ord_less_eq_o @ X2 @ B ) ) ) ) ).
% Sup_le_iff
thf(fact_492_Sup__upper,axiom,
! [X: set_real,A: set_set_real] :
( ( member_set_real @ X @ A )
=> ( ord_less_eq_set_real @ X @ ( comple3096694443085538997t_real @ A ) ) ) ).
% Sup_upper
thf(fact_493_Sup__upper,axiom,
! [X: $o,A: set_o] :
( ( member_o @ X @ A )
=> ( ord_less_eq_o @ X @ ( complete_Sup_Sup_o @ A ) ) ) ).
% Sup_upper
thf(fact_494_Sup__least,axiom,
! [A: set_set_real,Z2: set_real] :
( ! [X3: set_real] :
( ( member_set_real @ X3 @ A )
=> ( ord_less_eq_set_real @ X3 @ Z2 ) )
=> ( ord_less_eq_set_real @ ( comple3096694443085538997t_real @ A ) @ Z2 ) ) ).
% Sup_least
thf(fact_495_Sup__least,axiom,
! [A: set_o,Z2: $o] :
( ! [X3: $o] :
( ( member_o @ X3 @ A )
=> ( ord_less_eq_o @ X3 @ Z2 ) )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ A ) @ Z2 ) ) ).
% Sup_least
thf(fact_496_Sup__mono,axiom,
! [A: set_set_real,B4: set_set_real] :
( ! [A4: set_real] :
( ( member_set_real @ A4 @ A )
=> ? [X4: set_real] :
( ( member_set_real @ X4 @ B4 )
& ( ord_less_eq_set_real @ A4 @ X4 ) ) )
=> ( ord_less_eq_set_real @ ( comple3096694443085538997t_real @ A ) @ ( comple3096694443085538997t_real @ B4 ) ) ) ).
% Sup_mono
thf(fact_497_Sup__mono,axiom,
! [A: set_o,B4: set_o] :
( ! [A4: $o] :
( ( member_o @ A4 @ A )
=> ? [X4: $o] :
( ( member_o @ X4 @ B4 )
& ( ord_less_eq_o @ A4 @ X4 ) ) )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ A ) @ ( complete_Sup_Sup_o @ B4 ) ) ) ).
% Sup_mono
thf(fact_498_Sup__eqI,axiom,
! [A: set_set_real,X: set_real] :
( ! [Y2: set_real] :
( ( member_set_real @ Y2 @ A )
=> ( ord_less_eq_set_real @ Y2 @ X ) )
=> ( ! [Y2: set_real] :
( ! [Z3: set_real] :
( ( member_set_real @ Z3 @ A )
=> ( ord_less_eq_set_real @ Z3 @ Y2 ) )
=> ( ord_less_eq_set_real @ X @ Y2 ) )
=> ( ( comple3096694443085538997t_real @ A )
= X ) ) ) ).
% Sup_eqI
thf(fact_499_Sup__eqI,axiom,
! [A: set_o,X: $o] :
( ! [Y2: $o] :
( ( member_o @ Y2 @ A )
=> ( ord_less_eq_o @ Y2 @ X ) )
=> ( ! [Y2: $o] :
( ! [Z3: $o] :
( ( member_o @ Z3 @ A )
=> ( ord_less_eq_o @ Z3 @ Y2 ) )
=> ( ord_less_eq_o @ X @ Y2 ) )
=> ( ( complete_Sup_Sup_o @ A )
= X ) ) ) ).
% Sup_eqI
thf(fact_500_Inf__superset__mono,axiom,
! [B4: set_set_real,A: set_set_real] :
( ( ord_le3558479182127378552t_real @ B4 @ A )
=> ( ord_less_eq_set_real @ ( comple8289635161444856091t_real @ A ) @ ( comple8289635161444856091t_real @ B4 ) ) ) ).
% Inf_superset_mono
thf(fact_501_Inf__superset__mono,axiom,
! [B4: set_o,A: set_o] :
( ( ord_less_eq_set_o @ B4 @ A )
=> ( ord_less_eq_o @ ( complete_Inf_Inf_o @ A ) @ ( complete_Inf_Inf_o @ B4 ) ) ) ).
% Inf_superset_mono
thf(fact_502_Inf__greatest,axiom,
! [A: set_set_real,Z2: set_real] :
( ! [X3: set_real] :
( ( member_set_real @ X3 @ A )
=> ( ord_less_eq_set_real @ Z2 @ X3 ) )
=> ( ord_less_eq_set_real @ Z2 @ ( comple8289635161444856091t_real @ A ) ) ) ).
% Inf_greatest
thf(fact_503_Inf__greatest,axiom,
! [A: set_o,Z2: $o] :
( ! [X3: $o] :
( ( member_o @ X3 @ A )
=> ( ord_less_eq_o @ Z2 @ X3 ) )
=> ( ord_less_eq_o @ Z2 @ ( complete_Inf_Inf_o @ A ) ) ) ).
% Inf_greatest
thf(fact_504_le__Inf__iff,axiom,
! [B: set_real,A: set_set_real] :
( ( ord_less_eq_set_real @ B @ ( comple8289635161444856091t_real @ A ) )
= ( ! [X2: set_real] :
( ( member_set_real @ X2 @ A )
=> ( ord_less_eq_set_real @ B @ X2 ) ) ) ) ).
% le_Inf_iff
thf(fact_505_le__Inf__iff,axiom,
! [B: $o,A: set_o] :
( ( ord_less_eq_o @ B @ ( complete_Inf_Inf_o @ A ) )
= ( ! [X2: $o] :
( ( member_o @ X2 @ A )
=> ( ord_less_eq_o @ B @ X2 ) ) ) ) ).
% le_Inf_iff
thf(fact_506_Inf__lower2,axiom,
! [U2: set_real,A: set_set_real,V: set_real] :
( ( member_set_real @ U2 @ A )
=> ( ( ord_less_eq_set_real @ U2 @ V )
=> ( ord_less_eq_set_real @ ( comple8289635161444856091t_real @ A ) @ V ) ) ) ).
% Inf_lower2
thf(fact_507_Inf__lower2,axiom,
! [U2: $o,A: set_o,V: $o] :
( ( member_o @ U2 @ A )
=> ( ( ord_less_eq_o @ U2 @ V )
=> ( ord_less_eq_o @ ( complete_Inf_Inf_o @ A ) @ V ) ) ) ).
% Inf_lower2
thf(fact_508_Inf__lower,axiom,
! [X: set_real,A: set_set_real] :
( ( member_set_real @ X @ A )
=> ( ord_less_eq_set_real @ ( comple8289635161444856091t_real @ A ) @ X ) ) ).
% Inf_lower
thf(fact_509_Inf__lower,axiom,
! [X: $o,A: set_o] :
( ( member_o @ X @ A )
=> ( ord_less_eq_o @ ( complete_Inf_Inf_o @ A ) @ X ) ) ).
% Inf_lower
thf(fact_510_Inf__mono,axiom,
! [B4: set_set_real,A: set_set_real] :
( ! [B3: set_real] :
( ( member_set_real @ B3 @ B4 )
=> ? [X4: set_real] :
( ( member_set_real @ X4 @ A )
& ( ord_less_eq_set_real @ X4 @ B3 ) ) )
=> ( ord_less_eq_set_real @ ( comple8289635161444856091t_real @ A ) @ ( comple8289635161444856091t_real @ B4 ) ) ) ).
% Inf_mono
thf(fact_511_Inf__mono,axiom,
! [B4: set_o,A: set_o] :
( ! [B3: $o] :
( ( member_o @ B3 @ B4 )
=> ? [X4: $o] :
( ( member_o @ X4 @ A )
& ( ord_less_eq_o @ X4 @ B3 ) ) )
=> ( ord_less_eq_o @ ( complete_Inf_Inf_o @ A ) @ ( complete_Inf_Inf_o @ B4 ) ) ) ).
% Inf_mono
thf(fact_512_Inf__eqI,axiom,
! [A: set_set_real,X: set_real] :
( ! [I3: set_real] :
( ( member_set_real @ I3 @ A )
=> ( ord_less_eq_set_real @ X @ I3 ) )
=> ( ! [Y2: set_real] :
( ! [I4: set_real] :
( ( member_set_real @ I4 @ A )
=> ( ord_less_eq_set_real @ Y2 @ I4 ) )
=> ( ord_less_eq_set_real @ Y2 @ X ) )
=> ( ( comple8289635161444856091t_real @ A )
= X ) ) ) ).
% Inf_eqI
thf(fact_513_Inf__eqI,axiom,
! [A: set_o,X: $o] :
( ! [I3: $o] :
( ( member_o @ I3 @ A )
=> ( ord_less_eq_o @ X @ I3 ) )
=> ( ! [Y2: $o] :
( ! [I4: $o] :
( ( member_o @ I4 @ A )
=> ( ord_less_eq_o @ Y2 @ I4 ) )
=> ( ord_less_eq_o @ Y2 @ X ) )
=> ( ( complete_Inf_Inf_o @ A )
= X ) ) ) ).
% Inf_eqI
thf(fact_514_has__integral__on__superset,axiom,
! [F: real > real,I2: real,S: set_real,T: set_real] :
( ( hensto240673015341029504l_real @ F @ I2 @ S )
=> ( ! [X3: real] :
( ~ ( member_real @ X3 @ S )
=> ( ( F @ X3 )
= zero_zero_real ) )
=> ( ( ord_less_eq_set_real @ S @ T )
=> ( hensto240673015341029504l_real @ F @ I2 @ T ) ) ) ) ).
% has_integral_on_superset
thf(fact_515_has__integral__is__0,axiom,
! [S: set_real,F: real > real] :
( ! [X3: real] :
( ( member_real @ X3 @ S )
=> ( ( F @ X3 )
= zero_zero_real ) )
=> ( hensto240673015341029504l_real @ F @ zero_zero_real @ S ) ) ).
% has_integral_is_0
thf(fact_516_has__integral__le,axiom,
! [F: real > real,I2: real,S: set_real,G: real > real,J: real] :
( ( hensto240673015341029504l_real @ F @ I2 @ S )
=> ( ( hensto240673015341029504l_real @ G @ J @ S )
=> ( ! [X3: real] :
( ( member_real @ X3 @ S )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_less_eq_real @ I2 @ J ) ) ) ) ).
% has_integral_le
thf(fact_517_has__integral__0,axiom,
! [S: set_real] :
( hensto240673015341029504l_real
@ ^ [X2: real] : zero_zero_real
@ zero_zero_real
@ S ) ).
% has_integral_0
thf(fact_518_has__integral__mult__right,axiom,
! [F: real > real,Y: real,I2: set_real,C: real] :
( ( hensto240673015341029504l_real @ F @ Y @ I2 )
=> ( hensto240673015341029504l_real
@ ^ [X2: real] : ( times_times_real @ C @ ( F @ X2 ) )
@ ( times_times_real @ C @ Y )
@ I2 ) ) ).
% has_integral_mult_right
thf(fact_519_has__integral__mult__left,axiom,
! [F: real > real,Y: real,S: set_real,C: real] :
( ( hensto240673015341029504l_real @ F @ Y @ S )
=> ( hensto240673015341029504l_real
@ ^ [X2: real] : ( times_times_real @ ( F @ X2 ) @ C )
@ ( times_times_real @ Y @ C )
@ S ) ) ).
% has_integral_mult_left
thf(fact_520_has__integral__diff,axiom,
! [F: real > real,K: real,S: set_real,G: real > real,L: real] :
( ( hensto240673015341029504l_real @ F @ K @ S )
=> ( ( hensto240673015341029504l_real @ G @ L @ S )
=> ( hensto240673015341029504l_real
@ ^ [X2: real] : ( minus_minus_real @ ( F @ X2 ) @ ( G @ X2 ) )
@ ( minus_minus_real @ K @ L )
@ S ) ) ) ).
% has_integral_diff
thf(fact_521_has__integral__divide,axiom,
! [F: real > real,Y: real,S: set_real,C: real] :
( ( hensto240673015341029504l_real @ F @ Y @ S )
=> ( hensto240673015341029504l_real
@ ^ [X2: real] : ( divide_divide_real @ ( F @ X2 ) @ C )
@ ( divide_divide_real @ Y @ C )
@ S ) ) ).
% has_integral_divide
thf(fact_522_less__eq__Sup,axiom,
! [A: set_set_real,U2: set_real] :
( ! [V2: set_real] :
( ( member_set_real @ V2 @ A )
=> ( ord_less_eq_set_real @ U2 @ V2 ) )
=> ( ( A != bot_bot_set_set_real )
=> ( ord_less_eq_set_real @ U2 @ ( comple3096694443085538997t_real @ A ) ) ) ) ).
% less_eq_Sup
thf(fact_523_less__eq__Sup,axiom,
! [A: set_o,U2: $o] :
( ! [V2: $o] :
( ( member_o @ V2 @ A )
=> ( ord_less_eq_o @ U2 @ V2 ) )
=> ( ( A != bot_bot_set_o )
=> ( ord_less_eq_o @ U2 @ ( complete_Sup_Sup_o @ A ) ) ) ) ).
% less_eq_Sup
thf(fact_524_Inf__less__eq,axiom,
! [A: set_set_real,U2: set_real] :
( ! [V2: set_real] :
( ( member_set_real @ V2 @ A )
=> ( ord_less_eq_set_real @ V2 @ U2 ) )
=> ( ( A != bot_bot_set_set_real )
=> ( ord_less_eq_set_real @ ( comple8289635161444856091t_real @ A ) @ U2 ) ) ) ).
% Inf_less_eq
thf(fact_525_Inf__less__eq,axiom,
! [A: set_o,U2: $o] :
( ! [V2: $o] :
( ( member_o @ V2 @ A )
=> ( ord_less_eq_o @ V2 @ U2 ) )
=> ( ( A != bot_bot_set_o )
=> ( ord_less_eq_o @ ( complete_Inf_Inf_o @ A ) @ U2 ) ) ) ).
% Inf_less_eq
thf(fact_526_has__integral__subset__le,axiom,
! [S2: set_real,T2: set_real,F: real > real,I2: real,J: real] :
( ( ord_less_eq_set_real @ S2 @ T2 )
=> ( ( hensto240673015341029504l_real @ F @ I2 @ S2 )
=> ( ( hensto240673015341029504l_real @ F @ J @ T2 )
=> ( ! [X3: real] :
( ( member_real @ X3 @ T2 )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) )
=> ( ord_less_eq_real @ I2 @ J ) ) ) ) ) ).
% has_integral_subset_le
thf(fact_527_has__integral__nonneg,axiom,
! [F: real > real,I2: real,S: set_real] :
( ( hensto240673015341029504l_real @ F @ I2 @ S )
=> ( ! [X3: real] :
( ( member_real @ X3 @ S )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) )
=> ( ord_less_eq_real @ zero_zero_real @ I2 ) ) ) ).
% has_integral_nonneg
thf(fact_528_has__integral__cmult__real,axiom,
! [C: real,F: real > real,X: real,A: set_real] :
( ( ( C != zero_zero_real )
=> ( hensto240673015341029504l_real @ F @ X @ A ) )
=> ( hensto240673015341029504l_real
@ ^ [X2: real] : ( times_times_real @ C @ ( F @ X2 ) )
@ ( times_times_real @ C @ X )
@ A ) ) ).
% has_integral_cmult_real
thf(fact_529_Inf__le__Sup,axiom,
! [A: set_set_real] :
( ( A != bot_bot_set_set_real )
=> ( ord_less_eq_set_real @ ( comple8289635161444856091t_real @ A ) @ ( comple3096694443085538997t_real @ A ) ) ) ).
% Inf_le_Sup
thf(fact_530_Inf__le__Sup,axiom,
! [A: set_o] :
( ( A != bot_bot_set_o )
=> ( ord_less_eq_o @ ( complete_Inf_Inf_o @ A ) @ ( complete_Sup_Sup_o @ A ) ) ) ).
% Inf_le_Sup
thf(fact_531_cInf__atLeastAtMost,axiom,
! [Y: set_real,X: set_real] :
( ( ord_less_eq_set_real @ Y @ X )
=> ( ( comple8289635161444856091t_real @ ( set_or7743017856606604397t_real @ Y @ X ) )
= Y ) ) ).
% cInf_atLeastAtMost
thf(fact_532_cInf__atLeastAtMost,axiom,
! [Y: $o,X: $o] :
( ( ord_less_eq_o @ Y @ X )
=> ( ( complete_Inf_Inf_o @ ( set_or8904488021354931149Most_o @ Y @ X ) )
= Y ) ) ).
% cInf_atLeastAtMost
thf(fact_533_cInf__atLeastAtMost,axiom,
! [Y: real,X: real] :
( ( ord_less_eq_real @ Y @ X )
=> ( ( comple4887499456419720421f_real @ ( set_or1222579329274155063t_real @ Y @ X ) )
= Y ) ) ).
% cInf_atLeastAtMost
thf(fact_534_cInf__atLeastAtMost,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( complete_Inf_Inf_nat @ ( set_or1269000886237332187st_nat @ Y @ X ) )
= Y ) ) ).
% cInf_atLeastAtMost
thf(fact_535_cInf__atLeastAtMost,axiom,
! [Y: int,X: int] :
( ( ord_less_eq_int @ Y @ X )
=> ( ( complete_Inf_Inf_int @ ( set_or1266510415728281911st_int @ Y @ X ) )
= Y ) ) ).
% cInf_atLeastAtMost
thf(fact_536_Inf__atLeastAtMost,axiom,
! [X: set_real,Y: set_real] :
( ( ord_less_eq_set_real @ X @ Y )
=> ( ( comple8289635161444856091t_real @ ( set_or7743017856606604397t_real @ X @ Y ) )
= X ) ) ).
% Inf_atLeastAtMost
thf(fact_537_Inf__atLeastAtMost,axiom,
! [X: $o,Y: $o] :
( ( ord_less_eq_o @ X @ Y )
=> ( ( complete_Inf_Inf_o @ ( set_or8904488021354931149Most_o @ X @ Y ) )
= X ) ) ).
% Inf_atLeastAtMost
thf(fact_538_cSup__atLeastAtMost,axiom,
! [Y: set_real,X: set_real] :
( ( ord_less_eq_set_real @ Y @ X )
=> ( ( comple3096694443085538997t_real @ ( set_or7743017856606604397t_real @ Y @ X ) )
= X ) ) ).
% cSup_atLeastAtMost
thf(fact_539_cSup__atLeastAtMost,axiom,
! [Y: $o,X: $o] :
( ( ord_less_eq_o @ Y @ X )
=> ( ( complete_Sup_Sup_o @ ( set_or8904488021354931149Most_o @ Y @ X ) )
= X ) ) ).
% cSup_atLeastAtMost
thf(fact_540_cSup__atLeastAtMost,axiom,
! [Y: real,X: real] :
( ( ord_less_eq_real @ Y @ X )
=> ( ( comple1385675409528146559p_real @ ( set_or1222579329274155063t_real @ Y @ X ) )
= X ) ) ).
% cSup_atLeastAtMost
thf(fact_541_cSup__atLeastAtMost,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( complete_Sup_Sup_nat @ ( set_or1269000886237332187st_nat @ Y @ X ) )
= X ) ) ).
% cSup_atLeastAtMost
thf(fact_542_cSup__atLeastAtMost,axiom,
! [Y: int,X: int] :
( ( ord_less_eq_int @ Y @ X )
=> ( ( complete_Sup_Sup_int @ ( set_or1266510415728281911st_int @ Y @ X ) )
= X ) ) ).
% cSup_atLeastAtMost
thf(fact_543_Sup__atLeastAtMost,axiom,
! [X: set_real,Y: set_real] :
( ( ord_less_eq_set_real @ X @ Y )
=> ( ( comple3096694443085538997t_real @ ( set_or7743017856606604397t_real @ X @ Y ) )
= Y ) ) ).
% Sup_atLeastAtMost
thf(fact_544_Sup__atLeastAtMost,axiom,
! [X: $o,Y: $o] :
( ( ord_less_eq_o @ X @ Y )
=> ( ( complete_Sup_Sup_o @ ( set_or8904488021354931149Most_o @ X @ Y ) )
= Y ) ) ).
% Sup_atLeastAtMost
thf(fact_545_atLeastatMost__empty_H,axiom,
! [A2: set_real,B: set_real] :
( ~ ( ord_less_eq_set_real @ A2 @ B )
=> ( ( set_or7743017856606604397t_real @ A2 @ B )
= bot_bot_set_set_real ) ) ).
% atLeastatMost_empty'
thf(fact_546_atLeastatMost__empty_H,axiom,
! [A2: real,B: real] :
( ~ ( ord_less_eq_real @ A2 @ B )
=> ( ( set_or1222579329274155063t_real @ A2 @ B )
= bot_bot_set_real ) ) ).
% atLeastatMost_empty'
thf(fact_547_atLeastatMost__empty_H,axiom,
! [A2: nat,B: nat] :
( ~ ( ord_less_eq_nat @ A2 @ B )
=> ( ( set_or1269000886237332187st_nat @ A2 @ B )
= bot_bot_set_nat ) ) ).
% atLeastatMost_empty'
thf(fact_548_atLeastatMost__empty_H,axiom,
! [A2: int,B: int] :
( ~ ( ord_less_eq_int @ A2 @ B )
=> ( ( set_or1266510415728281911st_int @ A2 @ B )
= bot_bot_set_int ) ) ).
% atLeastatMost_empty'
thf(fact_549_atLeastatMost__empty__iff2,axiom,
! [A2: set_real,B: set_real] :
( ( bot_bot_set_set_real
= ( set_or7743017856606604397t_real @ A2 @ B ) )
= ( ~ ( ord_less_eq_set_real @ A2 @ B ) ) ) ).
% atLeastatMost_empty_iff2
thf(fact_550_atLeastatMost__empty__iff2,axiom,
! [A2: real,B: real] :
( ( bot_bot_set_real
= ( set_or1222579329274155063t_real @ A2 @ B ) )
= ( ~ ( ord_less_eq_real @ A2 @ B ) ) ) ).
% atLeastatMost_empty_iff2
thf(fact_551_atLeastatMost__empty__iff2,axiom,
! [A2: nat,B: nat] :
( ( bot_bot_set_nat
= ( set_or1269000886237332187st_nat @ A2 @ B ) )
= ( ~ ( ord_less_eq_nat @ A2 @ B ) ) ) ).
% atLeastatMost_empty_iff2
thf(fact_552_atLeastatMost__empty__iff2,axiom,
! [A2: int,B: int] :
( ( bot_bot_set_int
= ( set_or1266510415728281911st_int @ A2 @ B ) )
= ( ~ ( ord_less_eq_int @ A2 @ B ) ) ) ).
% atLeastatMost_empty_iff2
thf(fact_553_atLeastatMost__empty__iff,axiom,
! [A2: set_real,B: set_real] :
( ( ( set_or7743017856606604397t_real @ A2 @ B )
= bot_bot_set_set_real )
= ( ~ ( ord_less_eq_set_real @ A2 @ B ) ) ) ).
% atLeastatMost_empty_iff
thf(fact_554_atLeastatMost__empty__iff,axiom,
! [A2: real,B: real] :
( ( ( set_or1222579329274155063t_real @ A2 @ B )
= bot_bot_set_real )
= ( ~ ( ord_less_eq_real @ A2 @ B ) ) ) ).
% atLeastatMost_empty_iff
thf(fact_555_atLeastatMost__empty__iff,axiom,
! [A2: nat,B: nat] :
( ( ( set_or1269000886237332187st_nat @ A2 @ B )
= bot_bot_set_nat )
= ( ~ ( ord_less_eq_nat @ A2 @ B ) ) ) ).
% atLeastatMost_empty_iff
thf(fact_556_atLeastatMost__empty__iff,axiom,
! [A2: int,B: int] :
( ( ( set_or1266510415728281911st_int @ A2 @ B )
= bot_bot_set_int )
= ( ~ ( ord_less_eq_int @ A2 @ B ) ) ) ).
% atLeastatMost_empty_iff
thf(fact_557_atLeastatMost__subset__iff,axiom,
! [A2: set_real,B: set_real,C: set_real,D: set_real] :
( ( ord_le3558479182127378552t_real @ ( set_or7743017856606604397t_real @ A2 @ B ) @ ( set_or7743017856606604397t_real @ C @ D ) )
= ( ~ ( ord_less_eq_set_real @ A2 @ B )
| ( ( ord_less_eq_set_real @ C @ A2 )
& ( ord_less_eq_set_real @ B @ D ) ) ) ) ).
% atLeastatMost_subset_iff
thf(fact_558_atLeastatMost__subset__iff,axiom,
! [A2: real,B: real,C: real,D: real] :
( ( ord_less_eq_set_real @ ( set_or1222579329274155063t_real @ A2 @ B ) @ ( set_or1222579329274155063t_real @ C @ D ) )
= ( ~ ( ord_less_eq_real @ A2 @ B )
| ( ( ord_less_eq_real @ C @ A2 )
& ( ord_less_eq_real @ B @ D ) ) ) ) ).
% atLeastatMost_subset_iff
thf(fact_559_atLeastatMost__subset__iff,axiom,
! [A2: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_set_nat @ ( set_or1269000886237332187st_nat @ A2 @ B ) @ ( set_or1269000886237332187st_nat @ C @ D ) )
= ( ~ ( ord_less_eq_nat @ A2 @ B )
| ( ( ord_less_eq_nat @ C @ A2 )
& ( ord_less_eq_nat @ B @ D ) ) ) ) ).
% atLeastatMost_subset_iff
thf(fact_560_atLeastatMost__subset__iff,axiom,
! [A2: int,B: int,C: int,D: int] :
( ( ord_less_eq_set_int @ ( set_or1266510415728281911st_int @ A2 @ B ) @ ( set_or1266510415728281911st_int @ C @ D ) )
= ( ~ ( ord_less_eq_int @ A2 @ B )
| ( ( ord_less_eq_int @ C @ A2 )
& ( ord_less_eq_int @ B @ D ) ) ) ) ).
% atLeastatMost_subset_iff
thf(fact_561_Sup__nat__empty,axiom,
( ( complete_Sup_Sup_nat @ bot_bot_set_nat )
= zero_zero_nat ) ).
% Sup_nat_empty
thf(fact_562_UN__ball__bex__simps_I3_J,axiom,
! [A: set_set_real,P: real > $o] :
( ( ? [X2: real] :
( ( member_real @ X2 @ ( comple3096694443085538997t_real @ A ) )
& ( P @ X2 ) ) )
= ( ? [X2: set_real] :
( ( member_set_real @ X2 @ A )
& ? [Y5: real] :
( ( member_real @ Y5 @ X2 )
& ( P @ Y5 ) ) ) ) ) ).
% UN_ball_bex_simps(3)
thf(fact_563_UN__ball__bex__simps_I1_J,axiom,
! [A: set_set_real,P: real > $o] :
( ( ! [X2: real] :
( ( member_real @ X2 @ ( comple3096694443085538997t_real @ A ) )
=> ( P @ X2 ) ) )
= ( ! [X2: set_real] :
( ( member_set_real @ X2 @ A )
=> ! [Y5: real] :
( ( member_real @ Y5 @ X2 )
=> ( P @ Y5 ) ) ) ) ) ).
% UN_ball_bex_simps(1)
thf(fact_564_UnionI,axiom,
! [X6: set_set_real,C2: set_set_set_real,A: set_real] :
( ( member_set_set_real @ X6 @ C2 )
=> ( ( member_set_real @ A @ X6 )
=> ( member_set_real @ A @ ( comple5917660045593844715t_real @ C2 ) ) ) ) ).
% UnionI
thf(fact_565_UnionI,axiom,
! [X6: set_nat,C2: set_set_nat,A: nat] :
( ( member_set_nat @ X6 @ C2 )
=> ( ( member_nat @ A @ X6 )
=> ( member_nat @ A @ ( comple7399068483239264473et_nat @ C2 ) ) ) ) ).
% UnionI
thf(fact_566_UnionI,axiom,
! [X6: set_o,C2: set_set_o,A: $o] :
( ( member_set_o @ X6 @ C2 )
=> ( ( member_o @ A @ X6 )
=> ( member_o @ A @ ( comple90263536869209701_set_o @ C2 ) ) ) ) ).
% UnionI
thf(fact_567_UnionI,axiom,
! [X6: set_real,C2: set_set_real,A: real] :
( ( member_set_real @ X6 @ C2 )
=> ( ( member_real @ A @ X6 )
=> ( member_real @ A @ ( comple3096694443085538997t_real @ C2 ) ) ) ) ).
% UnionI
thf(fact_568_Union__iff,axiom,
! [A: set_real,C2: set_set_set_real] :
( ( member_set_real @ A @ ( comple5917660045593844715t_real @ C2 ) )
= ( ? [X2: set_set_real] :
( ( member_set_set_real @ X2 @ C2 )
& ( member_set_real @ A @ X2 ) ) ) ) ).
% Union_iff
thf(fact_569_Union__iff,axiom,
! [A: nat,C2: set_set_nat] :
( ( member_nat @ A @ ( comple7399068483239264473et_nat @ C2 ) )
= ( ? [X2: set_nat] :
( ( member_set_nat @ X2 @ C2 )
& ( member_nat @ A @ X2 ) ) ) ) ).
% Union_iff
thf(fact_570_Union__iff,axiom,
! [A: $o,C2: set_set_o] :
( ( member_o @ A @ ( comple90263536869209701_set_o @ C2 ) )
= ( ? [X2: set_o] :
( ( member_set_o @ X2 @ C2 )
& ( member_o @ A @ X2 ) ) ) ) ).
% Union_iff
thf(fact_571_Union__iff,axiom,
! [A: real,C2: set_set_real] :
( ( member_real @ A @ ( comple3096694443085538997t_real @ C2 ) )
= ( ? [X2: set_real] :
( ( member_set_real @ X2 @ C2 )
& ( member_real @ A @ X2 ) ) ) ) ).
% Union_iff
thf(fact_572_Icc__eq__Icc,axiom,
! [L: set_real,H: set_real,L2: set_real,H3: set_real] :
( ( ( set_or7743017856606604397t_real @ L @ H )
= ( set_or7743017856606604397t_real @ L2 @ H3 ) )
= ( ( ( L = L2 )
& ( H = H3 ) )
| ( ~ ( ord_less_eq_set_real @ L @ H )
& ~ ( ord_less_eq_set_real @ L2 @ H3 ) ) ) ) ).
% Icc_eq_Icc
thf(fact_573_Icc__eq__Icc,axiom,
! [L: real,H: real,L2: real,H3: real] :
( ( ( set_or1222579329274155063t_real @ L @ H )
= ( set_or1222579329274155063t_real @ L2 @ H3 ) )
= ( ( ( L = L2 )
& ( H = H3 ) )
| ( ~ ( ord_less_eq_real @ L @ H )
& ~ ( ord_less_eq_real @ L2 @ H3 ) ) ) ) ).
% Icc_eq_Icc
thf(fact_574_Icc__eq__Icc,axiom,
! [L: nat,H: nat,L2: nat,H3: nat] :
( ( ( set_or1269000886237332187st_nat @ L @ H )
= ( set_or1269000886237332187st_nat @ L2 @ H3 ) )
= ( ( ( L = L2 )
& ( H = H3 ) )
| ( ~ ( ord_less_eq_nat @ L @ H )
& ~ ( ord_less_eq_nat @ L2 @ H3 ) ) ) ) ).
% Icc_eq_Icc
thf(fact_575_Icc__eq__Icc,axiom,
! [L: int,H: int,L2: int,H3: int] :
( ( ( set_or1266510415728281911st_int @ L @ H )
= ( set_or1266510415728281911st_int @ L2 @ H3 ) )
= ( ( ( L = L2 )
& ( H = H3 ) )
| ( ~ ( ord_less_eq_int @ L @ H )
& ~ ( ord_less_eq_int @ L2 @ H3 ) ) ) ) ).
% Icc_eq_Icc
thf(fact_576_atLeastAtMost__iff,axiom,
! [I2: $o,L: $o,U2: $o] :
( ( member_o @ I2 @ ( set_or8904488021354931149Most_o @ L @ U2 ) )
= ( ( ord_less_eq_o @ L @ I2 )
& ( ord_less_eq_o @ I2 @ U2 ) ) ) ).
% atLeastAtMost_iff
thf(fact_577_atLeastAtMost__iff,axiom,
! [I2: set_real,L: set_real,U2: set_real] :
( ( member_set_real @ I2 @ ( set_or7743017856606604397t_real @ L @ U2 ) )
= ( ( ord_less_eq_set_real @ L @ I2 )
& ( ord_less_eq_set_real @ I2 @ U2 ) ) ) ).
% atLeastAtMost_iff
thf(fact_578_atLeastAtMost__iff,axiom,
! [I2: real,L: real,U2: real] :
( ( member_real @ I2 @ ( set_or1222579329274155063t_real @ L @ U2 ) )
= ( ( ord_less_eq_real @ L @ I2 )
& ( ord_less_eq_real @ I2 @ U2 ) ) ) ).
% atLeastAtMost_iff
thf(fact_579_atLeastAtMost__iff,axiom,
! [I2: nat,L: nat,U2: nat] :
( ( member_nat @ I2 @ ( set_or1269000886237332187st_nat @ L @ U2 ) )
= ( ( ord_less_eq_nat @ L @ I2 )
& ( ord_less_eq_nat @ I2 @ U2 ) ) ) ).
% atLeastAtMost_iff
thf(fact_580_atLeastAtMost__iff,axiom,
! [I2: int,L: int,U2: int] :
( ( member_int @ I2 @ ( set_or1266510415728281911st_int @ L @ U2 ) )
= ( ( ord_less_eq_int @ L @ I2 )
& ( ord_less_eq_int @ I2 @ U2 ) ) ) ).
% atLeastAtMost_iff
thf(fact_581_int__distrib_I4_J,axiom,
! [W: int,Z1: int,Z22: int] :
( ( times_times_int @ W @ ( minus_minus_int @ Z1 @ Z22 ) )
= ( minus_minus_int @ ( times_times_int @ W @ Z1 ) @ ( times_times_int @ W @ Z22 ) ) ) ).
% int_distrib(4)
thf(fact_582_int__distrib_I3_J,axiom,
! [Z1: int,Z22: int,W: int] :
( ( times_times_int @ ( minus_minus_int @ Z1 @ Z22 ) @ W )
= ( minus_minus_int @ ( times_times_int @ Z1 @ W ) @ ( times_times_int @ Z22 @ W ) ) ) ).
% int_distrib(3)
thf(fact_583_minus__int__code_I1_J,axiom,
! [K: int] :
( ( minus_minus_int @ K @ zero_zero_int )
= K ) ).
% minus_int_code(1)
thf(fact_584_times__int__code_I2_J,axiom,
! [L: int] :
( ( times_times_int @ zero_zero_int @ L )
= zero_zero_int ) ).
% times_int_code(2)
thf(fact_585_times__int__code_I1_J,axiom,
! [K: int] :
( ( times_times_int @ K @ zero_zero_int )
= zero_zero_int ) ).
% times_int_code(1)
thf(fact_586_less__eq__int__code_I1_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% less_eq_int_code(1)
thf(fact_587_zdiv__zmult2__eq,axiom,
! [C: int,A2: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ( divide_divide_int @ A2 @ ( times_times_int @ B @ C ) )
= ( divide_divide_int @ ( divide_divide_int @ A2 @ B ) @ C ) ) ) ).
% zdiv_zmult2_eq
thf(fact_588_UnionE,axiom,
! [A: set_real,C2: set_set_set_real] :
( ( member_set_real @ A @ ( comple5917660045593844715t_real @ C2 ) )
=> ~ ! [X5: set_set_real] :
( ( member_set_real @ A @ X5 )
=> ~ ( member_set_set_real @ X5 @ C2 ) ) ) ).
% UnionE
thf(fact_589_UnionE,axiom,
! [A: nat,C2: set_set_nat] :
( ( member_nat @ A @ ( comple7399068483239264473et_nat @ C2 ) )
=> ~ ! [X5: set_nat] :
( ( member_nat @ A @ X5 )
=> ~ ( member_set_nat @ X5 @ C2 ) ) ) ).
% UnionE
thf(fact_590_UnionE,axiom,
! [A: $o,C2: set_set_o] :
( ( member_o @ A @ ( comple90263536869209701_set_o @ C2 ) )
=> ~ ! [X5: set_o] :
( ( member_o @ A @ X5 )
=> ~ ( member_set_o @ X5 @ C2 ) ) ) ).
% UnionE
thf(fact_591_UnionE,axiom,
! [A: real,C2: set_set_real] :
( ( member_real @ A @ ( comple3096694443085538997t_real @ C2 ) )
=> ~ ! [X5: set_real] :
( ( member_real @ A @ X5 )
=> ~ ( member_set_real @ X5 @ C2 ) ) ) ).
% UnionE
thf(fact_592_Inter__lower,axiom,
! [B4: set_real,A: set_set_real] :
( ( member_set_real @ B4 @ A )
=> ( ord_less_eq_set_real @ ( comple8289635161444856091t_real @ A ) @ B4 ) ) ).
% Inter_lower
thf(fact_593_Inter__greatest,axiom,
! [A: set_set_real,C2: set_real] :
( ! [X5: set_real] :
( ( member_set_real @ X5 @ A )
=> ( ord_less_eq_set_real @ C2 @ X5 ) )
=> ( ord_less_eq_set_real @ C2 @ ( comple8289635161444856091t_real @ A ) ) ) ).
% Inter_greatest
thf(fact_594_Inter__anti__mono,axiom,
! [B4: set_set_real,A: set_set_real] :
( ( ord_le3558479182127378552t_real @ B4 @ A )
=> ( ord_less_eq_set_real @ ( comple8289635161444856091t_real @ A ) @ ( comple8289635161444856091t_real @ B4 ) ) ) ).
% Inter_anti_mono
thf(fact_595_Inter__subset,axiom,
! [A: set_set_real,B4: set_real] :
( ! [X5: set_real] :
( ( member_set_real @ X5 @ A )
=> ( ord_less_eq_set_real @ X5 @ B4 ) )
=> ( ( A != bot_bot_set_set_real )
=> ( ord_less_eq_set_real @ ( comple8289635161444856091t_real @ A ) @ B4 ) ) ) ).
% Inter_subset
thf(fact_596_bounded__Max__nat,axiom,
! [P: nat > $o,X: nat,M3: nat] :
( ( P @ X )
=> ( ! [X3: nat] :
( ( P @ X3 )
=> ( ord_less_eq_nat @ X3 @ M3 ) )
=> ~ ! [M2: nat] :
( ( P @ M2 )
=> ~ ! [X4: nat] :
( ( P @ X4 )
=> ( ord_less_eq_nat @ X4 @ M2 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_597_wellorder__InfI,axiom,
! [K: nat,A: set_nat] :
( ( member_nat @ K @ A )
=> ( member_nat @ ( complete_Inf_Inf_nat @ A ) @ A ) ) ).
% wellorder_InfI
thf(fact_598_cSup__eq__maximum,axiom,
! [Z2: int,X6: set_int] :
( ( member_int @ Z2 @ X6 )
=> ( ! [X3: int] :
( ( member_int @ X3 @ X6 )
=> ( ord_less_eq_int @ X3 @ Z2 ) )
=> ( ( complete_Sup_Sup_int @ X6 )
= Z2 ) ) ) ).
% cSup_eq_maximum
thf(fact_599_cSup__eq__maximum,axiom,
! [Z2: real,X6: set_real] :
( ( member_real @ Z2 @ X6 )
=> ( ! [X3: real] :
( ( member_real @ X3 @ X6 )
=> ( ord_less_eq_real @ X3 @ Z2 ) )
=> ( ( comple1385675409528146559p_real @ X6 )
= Z2 ) ) ) ).
% cSup_eq_maximum
thf(fact_600_cSup__eq__maximum,axiom,
! [Z2: set_real,X6: set_set_real] :
( ( member_set_real @ Z2 @ X6 )
=> ( ! [X3: set_real] :
( ( member_set_real @ X3 @ X6 )
=> ( ord_less_eq_set_real @ X3 @ Z2 ) )
=> ( ( comple3096694443085538997t_real @ X6 )
= Z2 ) ) ) ).
% cSup_eq_maximum
thf(fact_601_cSup__eq__maximum,axiom,
! [Z2: nat,X6: set_nat] :
( ( member_nat @ Z2 @ X6 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ X6 )
=> ( ord_less_eq_nat @ X3 @ Z2 ) )
=> ( ( complete_Sup_Sup_nat @ X6 )
= Z2 ) ) ) ).
% cSup_eq_maximum
thf(fact_602_cSup__eq__maximum,axiom,
! [Z2: $o,X6: set_o] :
( ( member_o @ Z2 @ X6 )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ X6 )
=> ( ord_less_eq_o @ X3 @ Z2 ) )
=> ( ( complete_Sup_Sup_o @ X6 )
= Z2 ) ) ) ).
% cSup_eq_maximum
thf(fact_603_cSup__eq,axiom,
! [X6: set_int,A2: int] :
( ! [X3: int] :
( ( member_int @ X3 @ X6 )
=> ( ord_less_eq_int @ X3 @ A2 ) )
=> ( ! [Y2: int] :
( ! [X4: int] :
( ( member_int @ X4 @ X6 )
=> ( ord_less_eq_int @ X4 @ Y2 ) )
=> ( ord_less_eq_int @ A2 @ Y2 ) )
=> ( ( complete_Sup_Sup_int @ X6 )
= A2 ) ) ) ).
% cSup_eq
thf(fact_604_cSup__eq,axiom,
! [X6: set_real,A2: real] :
( ! [X3: real] :
( ( member_real @ X3 @ X6 )
=> ( ord_less_eq_real @ X3 @ A2 ) )
=> ( ! [Y2: real] :
( ! [X4: real] :
( ( member_real @ X4 @ X6 )
=> ( ord_less_eq_real @ X4 @ Y2 ) )
=> ( ord_less_eq_real @ A2 @ Y2 ) )
=> ( ( comple1385675409528146559p_real @ X6 )
= A2 ) ) ) ).
% cSup_eq
thf(fact_605_cInf__eq__minimum,axiom,
! [Z2: int,X6: set_int] :
( ( member_int @ Z2 @ X6 )
=> ( ! [X3: int] :
( ( member_int @ X3 @ X6 )
=> ( ord_less_eq_int @ Z2 @ X3 ) )
=> ( ( complete_Inf_Inf_int @ X6 )
= Z2 ) ) ) ).
% cInf_eq_minimum
thf(fact_606_cInf__eq__minimum,axiom,
! [Z2: set_real,X6: set_set_real] :
( ( member_set_real @ Z2 @ X6 )
=> ( ! [X3: set_real] :
( ( member_set_real @ X3 @ X6 )
=> ( ord_less_eq_set_real @ Z2 @ X3 ) )
=> ( ( comple8289635161444856091t_real @ X6 )
= Z2 ) ) ) ).
% cInf_eq_minimum
thf(fact_607_cInf__eq__minimum,axiom,
! [Z2: real,X6: set_real] :
( ( member_real @ Z2 @ X6 )
=> ( ! [X3: real] :
( ( member_real @ X3 @ X6 )
=> ( ord_less_eq_real @ Z2 @ X3 ) )
=> ( ( comple4887499456419720421f_real @ X6 )
= Z2 ) ) ) ).
% cInf_eq_minimum
thf(fact_608_cInf__eq__minimum,axiom,
! [Z2: nat,X6: set_nat] :
( ( member_nat @ Z2 @ X6 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ X6 )
=> ( ord_less_eq_nat @ Z2 @ X3 ) )
=> ( ( complete_Inf_Inf_nat @ X6 )
= Z2 ) ) ) ).
% cInf_eq_minimum
thf(fact_609_cInf__eq__minimum,axiom,
! [Z2: $o,X6: set_o] :
( ( member_o @ Z2 @ X6 )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ X6 )
=> ( ord_less_eq_o @ Z2 @ X3 ) )
=> ( ( complete_Inf_Inf_o @ X6 )
= Z2 ) ) ) ).
% cInf_eq_minimum
thf(fact_610_cInf__eq,axiom,
! [X6: set_int,A2: int] :
( ! [X3: int] :
( ( member_int @ X3 @ X6 )
=> ( ord_less_eq_int @ A2 @ X3 ) )
=> ( ! [Y2: int] :
( ! [X4: int] :
( ( member_int @ X4 @ X6 )
=> ( ord_less_eq_int @ Y2 @ X4 ) )
=> ( ord_less_eq_int @ Y2 @ A2 ) )
=> ( ( complete_Inf_Inf_int @ X6 )
= A2 ) ) ) ).
% cInf_eq
thf(fact_611_cInf__eq,axiom,
! [X6: set_real,A2: real] :
( ! [X3: real] :
( ( member_real @ X3 @ X6 )
=> ( ord_less_eq_real @ A2 @ X3 ) )
=> ( ! [Y2: real] :
( ! [X4: real] :
( ( member_real @ X4 @ X6 )
=> ( ord_less_eq_real @ Y2 @ X4 ) )
=> ( ord_less_eq_real @ Y2 @ A2 ) )
=> ( ( comple4887499456419720421f_real @ X6 )
= A2 ) ) ) ).
% cInf_eq
thf(fact_612_cInf__eq,axiom,
! [X6: set_nat,A2: nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ X6 )
=> ( ord_less_eq_nat @ A2 @ X3 ) )
=> ( ! [Y2: nat] :
( ! [X4: nat] :
( ( member_nat @ X4 @ X6 )
=> ( ord_less_eq_nat @ Y2 @ X4 ) )
=> ( ord_less_eq_nat @ Y2 @ A2 ) )
=> ( ( complete_Inf_Inf_nat @ X6 )
= A2 ) ) ) ).
% cInf_eq
thf(fact_613_wellorder__Inf__le1,axiom,
! [K: nat,A: set_nat] :
( ( member_nat @ K @ A )
=> ( ord_less_eq_nat @ ( complete_Inf_Inf_nat @ A ) @ K ) ) ).
% wellorder_Inf_le1
thf(fact_614_cSup__least,axiom,
! [X6: set_int,Z2: int] :
( ( X6 != bot_bot_set_int )
=> ( ! [X3: int] :
( ( member_int @ X3 @ X6 )
=> ( ord_less_eq_int @ X3 @ Z2 ) )
=> ( ord_less_eq_int @ ( complete_Sup_Sup_int @ X6 ) @ Z2 ) ) ) ).
% cSup_least
thf(fact_615_cSup__least,axiom,
! [X6: set_real,Z2: real] :
( ( X6 != bot_bot_set_real )
=> ( ! [X3: real] :
( ( member_real @ X3 @ X6 )
=> ( ord_less_eq_real @ X3 @ Z2 ) )
=> ( ord_less_eq_real @ ( comple1385675409528146559p_real @ X6 ) @ Z2 ) ) ) ).
% cSup_least
thf(fact_616_cSup__least,axiom,
! [X6: set_set_real,Z2: set_real] :
( ( X6 != bot_bot_set_set_real )
=> ( ! [X3: set_real] :
( ( member_set_real @ X3 @ X6 )
=> ( ord_less_eq_set_real @ X3 @ Z2 ) )
=> ( ord_less_eq_set_real @ ( comple3096694443085538997t_real @ X6 ) @ Z2 ) ) ) ).
% cSup_least
thf(fact_617_cSup__least,axiom,
! [X6: set_nat,Z2: nat] :
( ( X6 != bot_bot_set_nat )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ X6 )
=> ( ord_less_eq_nat @ X3 @ Z2 ) )
=> ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ X6 ) @ Z2 ) ) ) ).
% cSup_least
thf(fact_618_cSup__least,axiom,
! [X6: set_o,Z2: $o] :
( ( X6 != bot_bot_set_o )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ X6 )
=> ( ord_less_eq_o @ X3 @ Z2 ) )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ X6 ) @ Z2 ) ) ) ).
% cSup_least
thf(fact_619_cSup__eq__non__empty,axiom,
! [X6: set_int,A2: int] :
( ( X6 != bot_bot_set_int )
=> ( ! [X3: int] :
( ( member_int @ X3 @ X6 )
=> ( ord_less_eq_int @ X3 @ A2 ) )
=> ( ! [Y2: int] :
( ! [X4: int] :
( ( member_int @ X4 @ X6 )
=> ( ord_less_eq_int @ X4 @ Y2 ) )
=> ( ord_less_eq_int @ A2 @ Y2 ) )
=> ( ( complete_Sup_Sup_int @ X6 )
= A2 ) ) ) ) ).
% cSup_eq_non_empty
thf(fact_620_cSup__eq__non__empty,axiom,
! [X6: set_real,A2: real] :
( ( X6 != bot_bot_set_real )
=> ( ! [X3: real] :
( ( member_real @ X3 @ X6 )
=> ( ord_less_eq_real @ X3 @ A2 ) )
=> ( ! [Y2: real] :
( ! [X4: real] :
( ( member_real @ X4 @ X6 )
=> ( ord_less_eq_real @ X4 @ Y2 ) )
=> ( ord_less_eq_real @ A2 @ Y2 ) )
=> ( ( comple1385675409528146559p_real @ X6 )
= A2 ) ) ) ) ).
% cSup_eq_non_empty
thf(fact_621_cSup__eq__non__empty,axiom,
! [X6: set_set_real,A2: set_real] :
( ( X6 != bot_bot_set_set_real )
=> ( ! [X3: set_real] :
( ( member_set_real @ X3 @ X6 )
=> ( ord_less_eq_set_real @ X3 @ A2 ) )
=> ( ! [Y2: set_real] :
( ! [X4: set_real] :
( ( member_set_real @ X4 @ X6 )
=> ( ord_less_eq_set_real @ X4 @ Y2 ) )
=> ( ord_less_eq_set_real @ A2 @ Y2 ) )
=> ( ( comple3096694443085538997t_real @ X6 )
= A2 ) ) ) ) ).
% cSup_eq_non_empty
thf(fact_622_cSup__eq__non__empty,axiom,
! [X6: set_nat,A2: nat] :
( ( X6 != bot_bot_set_nat )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ X6 )
=> ( ord_less_eq_nat @ X3 @ A2 ) )
=> ( ! [Y2: nat] :
( ! [X4: nat] :
( ( member_nat @ X4 @ X6 )
=> ( ord_less_eq_nat @ X4 @ Y2 ) )
=> ( ord_less_eq_nat @ A2 @ Y2 ) )
=> ( ( complete_Sup_Sup_nat @ X6 )
= A2 ) ) ) ) ).
% cSup_eq_non_empty
thf(fact_623_cSup__eq__non__empty,axiom,
! [X6: set_o,A2: $o] :
( ( X6 != bot_bot_set_o )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ X6 )
=> ( ord_less_eq_o @ X3 @ A2 ) )
=> ( ! [Y2: $o] :
( ! [X4: $o] :
( ( member_o @ X4 @ X6 )
=> ( ord_less_eq_o @ X4 @ Y2 ) )
=> ( ord_less_eq_o @ A2 @ Y2 ) )
=> ( ( complete_Sup_Sup_o @ X6 )
= A2 ) ) ) ) ).
% cSup_eq_non_empty
thf(fact_624_cInf__greatest,axiom,
! [X6: set_int,Z2: int] :
( ( X6 != bot_bot_set_int )
=> ( ! [X3: int] :
( ( member_int @ X3 @ X6 )
=> ( ord_less_eq_int @ Z2 @ X3 ) )
=> ( ord_less_eq_int @ Z2 @ ( complete_Inf_Inf_int @ X6 ) ) ) ) ).
% cInf_greatest
thf(fact_625_cInf__greatest,axiom,
! [X6: set_set_real,Z2: set_real] :
( ( X6 != bot_bot_set_set_real )
=> ( ! [X3: set_real] :
( ( member_set_real @ X3 @ X6 )
=> ( ord_less_eq_set_real @ Z2 @ X3 ) )
=> ( ord_less_eq_set_real @ Z2 @ ( comple8289635161444856091t_real @ X6 ) ) ) ) ).
% cInf_greatest
thf(fact_626_cInf__greatest,axiom,
! [X6: set_real,Z2: real] :
( ( X6 != bot_bot_set_real )
=> ( ! [X3: real] :
( ( member_real @ X3 @ X6 )
=> ( ord_less_eq_real @ Z2 @ X3 ) )
=> ( ord_less_eq_real @ Z2 @ ( comple4887499456419720421f_real @ X6 ) ) ) ) ).
% cInf_greatest
thf(fact_627_cInf__greatest,axiom,
! [X6: set_nat,Z2: nat] :
( ( X6 != bot_bot_set_nat )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ X6 )
=> ( ord_less_eq_nat @ Z2 @ X3 ) )
=> ( ord_less_eq_nat @ Z2 @ ( complete_Inf_Inf_nat @ X6 ) ) ) ) ).
% cInf_greatest
thf(fact_628_cInf__greatest,axiom,
! [X6: set_o,Z2: $o] :
( ( X6 != bot_bot_set_o )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ X6 )
=> ( ord_less_eq_o @ Z2 @ X3 ) )
=> ( ord_less_eq_o @ Z2 @ ( complete_Inf_Inf_o @ X6 ) ) ) ) ).
% cInf_greatest
thf(fact_629_cInf__eq__non__empty,axiom,
! [X6: set_int,A2: int] :
( ( X6 != bot_bot_set_int )
=> ( ! [X3: int] :
( ( member_int @ X3 @ X6 )
=> ( ord_less_eq_int @ A2 @ X3 ) )
=> ( ! [Y2: int] :
( ! [X4: int] :
( ( member_int @ X4 @ X6 )
=> ( ord_less_eq_int @ Y2 @ X4 ) )
=> ( ord_less_eq_int @ Y2 @ A2 ) )
=> ( ( complete_Inf_Inf_int @ X6 )
= A2 ) ) ) ) ).
% cInf_eq_non_empty
thf(fact_630_cInf__eq__non__empty,axiom,
! [X6: set_set_real,A2: set_real] :
( ( X6 != bot_bot_set_set_real )
=> ( ! [X3: set_real] :
( ( member_set_real @ X3 @ X6 )
=> ( ord_less_eq_set_real @ A2 @ X3 ) )
=> ( ! [Y2: set_real] :
( ! [X4: set_real] :
( ( member_set_real @ X4 @ X6 )
=> ( ord_less_eq_set_real @ Y2 @ X4 ) )
=> ( ord_less_eq_set_real @ Y2 @ A2 ) )
=> ( ( comple8289635161444856091t_real @ X6 )
= A2 ) ) ) ) ).
% cInf_eq_non_empty
thf(fact_631_cInf__eq__non__empty,axiom,
! [X6: set_real,A2: real] :
( ( X6 != bot_bot_set_real )
=> ( ! [X3: real] :
( ( member_real @ X3 @ X6 )
=> ( ord_less_eq_real @ A2 @ X3 ) )
=> ( ! [Y2: real] :
( ! [X4: real] :
( ( member_real @ X4 @ X6 )
=> ( ord_less_eq_real @ Y2 @ X4 ) )
=> ( ord_less_eq_real @ Y2 @ A2 ) )
=> ( ( comple4887499456419720421f_real @ X6 )
= A2 ) ) ) ) ).
% cInf_eq_non_empty
thf(fact_632_cInf__eq__non__empty,axiom,
! [X6: set_nat,A2: nat] :
( ( X6 != bot_bot_set_nat )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ X6 )
=> ( ord_less_eq_nat @ A2 @ X3 ) )
=> ( ! [Y2: nat] :
( ! [X4: nat] :
( ( member_nat @ X4 @ X6 )
=> ( ord_less_eq_nat @ Y2 @ X4 ) )
=> ( ord_less_eq_nat @ Y2 @ A2 ) )
=> ( ( complete_Inf_Inf_nat @ X6 )
= A2 ) ) ) ) ).
% cInf_eq_non_empty
thf(fact_633_cInf__eq__non__empty,axiom,
! [X6: set_o,A2: $o] :
( ( X6 != bot_bot_set_o )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ X6 )
=> ( ord_less_eq_o @ A2 @ X3 ) )
=> ( ! [Y2: $o] :
( ! [X4: $o] :
( ( member_o @ X4 @ X6 )
=> ( ord_less_eq_o @ Y2 @ X4 ) )
=> ( ord_less_eq_o @ Y2 @ A2 ) )
=> ( ( complete_Inf_Inf_o @ X6 )
= A2 ) ) ) ) ).
% cInf_eq_non_empty
thf(fact_634_cSup__abs__le,axiom,
! [S: set_int,A2: int] :
( ( S != bot_bot_set_int )
=> ( ! [X3: int] :
( ( member_int @ X3 @ S )
=> ( ord_less_eq_int @ ( abs_abs_int @ X3 ) @ A2 ) )
=> ( ord_less_eq_int @ ( abs_abs_int @ ( complete_Sup_Sup_int @ S ) ) @ A2 ) ) ) ).
% cSup_abs_le
thf(fact_635_cSup__abs__le,axiom,
! [S: set_real,A2: real] :
( ( S != bot_bot_set_real )
=> ( ! [X3: real] :
( ( member_real @ X3 @ S )
=> ( ord_less_eq_real @ ( abs_abs_real @ X3 ) @ A2 ) )
=> ( ord_less_eq_real @ ( abs_abs_real @ ( comple1385675409528146559p_real @ S ) ) @ A2 ) ) ) ).
% cSup_abs_le
thf(fact_636__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062del_O_A_092_060lbrakk_062_092_060And_062e_O_A0_A_060_Ae_A_092_060Longrightarrow_062_A0_A_060_Adel_Ae_059_A_092_060And_062e_Ax_Ax_H_O_A_092_060lbrakk_062_092_060bar_062x_H_A_N_Ax_092_060bar_062_A_060_Adel_Ae_059_A0_A_060_Ae_059_Ax_A_092_060in_062_A_1230_O_Oa_125_059_Ax_H_A_092_060in_062_A_1230_O_Oa_125_092_060rbrakk_062_A_092_060Longrightarrow_062_A_092_060bar_062f_Ax_H_A_N_Af_Ax_092_060bar_062_A_060_Ae_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [Del: real > real] :
( ! [E: real] :
( ( ord_less_real @ zero_zero_real @ E )
=> ( ord_less_real @ zero_zero_real @ ( Del @ E ) ) )
=> ~ ! [E: real,X4: real,X7: real] :
( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X7 @ X4 ) ) @ ( Del @ E ) )
=> ( ( ord_less_real @ zero_zero_real @ E )
=> ( ( member_real @ X4 @ ( set_or1222579329274155063t_real @ zero_zero_real @ a ) )
=> ( ( member_real @ X7 @ ( set_or1222579329274155063t_real @ zero_zero_real @ a ) )
=> ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ ( f @ X7 ) @ ( f @ X4 ) ) ) @ E ) ) ) ) ) ) ).
% \<open>\<And>thesis. (\<And>del. \<lbrakk>\<And>e. 0 < e \<Longrightarrow> 0 < del e; \<And>e x x'. \<lbrakk>\<bar>x' - x\<bar> < del e; 0 < e; x \<in> {0..a}; x' \<in> {0..a}\<rbrakk> \<Longrightarrow> \<bar>f x' - f x\<bar> < e\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_637_real__of__nat__div2,axiom,
! [N: nat,X: nat] : ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ X ) ) @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N @ X ) ) ) ) ).
% real_of_nat_div2
thf(fact_638_fim,axiom,
( ( image_real_real @ f @ ( set_or1222579329274155063t_real @ zero_zero_real @ a ) )
= ( set_or1222579329274155063t_real @ zero_zero_real @ b ) ) ).
% fim
thf(fact_639_cInf__asclose,axiom,
! [S: set_int,L: int,E2: int] :
( ( S != bot_bot_set_int )
=> ( ! [X3: int] :
( ( member_int @ X3 @ S )
=> ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ X3 @ L ) ) @ E2 ) )
=> ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( complete_Inf_Inf_int @ S ) @ L ) ) @ E2 ) ) ) ).
% cInf_asclose
thf(fact_640_cInf__asclose,axiom,
! [S: set_real,L: real,E2: real] :
( ( S != bot_bot_set_real )
=> ( ! [X3: real] :
( ( member_real @ X3 @ S )
=> ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ X3 @ L ) ) @ E2 ) )
=> ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( comple4887499456419720421f_real @ S ) @ L ) ) @ E2 ) ) ) ).
% cInf_asclose
thf(fact_641_cSup__asclose,axiom,
! [S: set_int,L: int,E2: int] :
( ( S != bot_bot_set_int )
=> ( ! [X3: int] :
( ( member_int @ X3 @ S )
=> ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ X3 @ L ) ) @ E2 ) )
=> ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( complete_Sup_Sup_int @ S ) @ L ) ) @ E2 ) ) ) ).
% cSup_asclose
thf(fact_642_cSup__asclose,axiom,
! [S: set_real,L: real,E2: real] :
( ( S != bot_bot_set_real )
=> ( ! [X3: real] :
( ( member_real @ X3 @ S )
=> ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ X3 @ L ) ) @ E2 ) )
=> ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( comple1385675409528146559p_real @ S ) @ L ) ) @ E2 ) ) ) ).
% cSup_asclose
thf(fact_643_subset__empty,axiom,
! [A: set_set_real] :
( ( ord_le3558479182127378552t_real @ A @ bot_bot_set_set_real )
= ( A = bot_bot_set_set_real ) ) ).
% subset_empty
thf(fact_644_subset__empty,axiom,
! [A: set_nat] :
( ( ord_less_eq_set_nat @ A @ bot_bot_set_nat )
= ( A = bot_bot_set_nat ) ) ).
% subset_empty
thf(fact_645_subset__empty,axiom,
! [A: set_real] :
( ( ord_less_eq_set_real @ A @ bot_bot_set_real )
= ( A = bot_bot_set_real ) ) ).
% subset_empty
thf(fact_646_that,axiom,
ord_less_real @ zero_zero_real @ epsilon ).
% that
thf(fact_647__092_060open_0620_A_060_Aa_092_060close_062,axiom,
ord_less_real @ zero_zero_real @ a ).
% \<open>0 < a\<close>
thf(fact_648__092_060open_0620_A_060_A_092_060delta_062_092_060close_062,axiom,
ord_less_real @ zero_zero_real @ delta ).
% \<open>0 < \<delta>\<close>
thf(fact_649_image__eqI,axiom,
! [B: real,F: real > real,X: real,A: set_real] :
( ( B
= ( F @ X ) )
=> ( ( member_real @ X @ A )
=> ( member_real @ B @ ( image_real_real @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_650_image__eqI,axiom,
! [B: nat,F: real > nat,X: real,A: set_real] :
( ( B
= ( F @ X ) )
=> ( ( member_real @ X @ A )
=> ( member_nat @ B @ ( image_real_nat @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_651_image__eqI,axiom,
! [B: $o,F: real > $o,X: real,A: set_real] :
( ( B
= ( F @ X ) )
=> ( ( member_real @ X @ A )
=> ( member_o @ B @ ( image_real_o @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_652_image__eqI,axiom,
! [B: int,F: nat > int,X: nat,A: set_nat] :
( ( B
= ( F @ X ) )
=> ( ( member_nat @ X @ A )
=> ( member_int @ B @ ( image_nat_int @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_653_image__eqI,axiom,
! [B: real,F: nat > real,X: nat,A: set_nat] :
( ( B
= ( F @ X ) )
=> ( ( member_nat @ X @ A )
=> ( member_real @ B @ ( image_nat_real @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_654_image__eqI,axiom,
! [B: nat,F: nat > nat,X: nat,A: set_nat] :
( ( B
= ( F @ X ) )
=> ( ( member_nat @ X @ A )
=> ( member_nat @ B @ ( image_nat_nat @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_655_image__eqI,axiom,
! [B: $o,F: nat > $o,X: nat,A: set_nat] :
( ( B
= ( F @ X ) )
=> ( ( member_nat @ X @ A )
=> ( member_o @ B @ ( image_nat_o @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_656_image__eqI,axiom,
! [B: real,F: $o > real,X: $o,A: set_o] :
( ( B
= ( F @ X ) )
=> ( ( member_o @ X @ A )
=> ( member_real @ B @ ( image_o_real @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_657_image__eqI,axiom,
! [B: nat,F: $o > nat,X: $o,A: set_o] :
( ( B
= ( F @ X ) )
=> ( ( member_o @ X @ A )
=> ( member_nat @ B @ ( image_o_nat @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_658_image__eqI,axiom,
! [B: $o,F: $o > $o,X: $o,A: set_o] :
( ( B
= ( F @ X ) )
=> ( ( member_o @ X @ A )
=> ( member_o @ B @ ( image_o_o @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_659_subset__antisym,axiom,
! [A: set_real,B4: set_real] :
( ( ord_less_eq_set_real @ A @ B4 )
=> ( ( ord_less_eq_set_real @ B4 @ A )
=> ( A = B4 ) ) ) ).
% subset_antisym
thf(fact_660_subsetI,axiom,
! [A: set_set_real,B4: set_set_real] :
( ! [X3: set_real] :
( ( member_set_real @ X3 @ A )
=> ( member_set_real @ X3 @ B4 ) )
=> ( ord_le3558479182127378552t_real @ A @ B4 ) ) ).
% subsetI
thf(fact_661_subsetI,axiom,
! [A: set_nat,B4: set_nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( member_nat @ X3 @ B4 ) )
=> ( ord_less_eq_set_nat @ A @ B4 ) ) ).
% subsetI
thf(fact_662_subsetI,axiom,
! [A: set_o,B4: set_o] :
( ! [X3: $o] :
( ( member_o @ X3 @ A )
=> ( member_o @ X3 @ B4 ) )
=> ( ord_less_eq_set_o @ A @ B4 ) ) ).
% subsetI
thf(fact_663_subsetI,axiom,
! [A: set_real,B4: set_real] :
( ! [X3: real] :
( ( member_real @ X3 @ A )
=> ( member_real @ X3 @ B4 ) )
=> ( ord_less_eq_set_real @ A @ B4 ) ) ).
% subsetI
thf(fact_664_empty__iff,axiom,
! [C: $o] :
~ ( member_o @ C @ bot_bot_set_o ) ).
% empty_iff
thf(fact_665_empty__iff,axiom,
! [C: set_real] :
~ ( member_set_real @ C @ bot_bot_set_set_real ) ).
% empty_iff
thf(fact_666_empty__iff,axiom,
! [C: real] :
~ ( member_real @ C @ bot_bot_set_real ) ).
% empty_iff
thf(fact_667_empty__iff,axiom,
! [C: nat] :
~ ( member_nat @ C @ bot_bot_set_nat ) ).
% empty_iff
thf(fact_668_Diff__empty,axiom,
! [A: set_set_real] :
( ( minus_5467046032205032049t_real @ A @ bot_bot_set_set_real )
= A ) ).
% Diff_empty
thf(fact_669_Diff__empty,axiom,
! [A: set_real] :
( ( minus_minus_set_real @ A @ bot_bot_set_real )
= A ) ).
% Diff_empty
thf(fact_670_Diff__empty,axiom,
! [A: set_nat] :
( ( minus_minus_set_nat @ A @ bot_bot_set_nat )
= A ) ).
% Diff_empty
thf(fact_671_empty__Diff,axiom,
! [A: set_set_real] :
( ( minus_5467046032205032049t_real @ bot_bot_set_set_real @ A )
= bot_bot_set_set_real ) ).
% empty_Diff
thf(fact_672_empty__Diff,axiom,
! [A: set_real] :
( ( minus_minus_set_real @ bot_bot_set_real @ A )
= bot_bot_set_real ) ).
% empty_Diff
thf(fact_673_empty__Diff,axiom,
! [A: set_nat] :
( ( minus_minus_set_nat @ bot_bot_set_nat @ A )
= bot_bot_set_nat ) ).
% empty_Diff
thf(fact_674_Diff__cancel,axiom,
! [A: set_set_real] :
( ( minus_5467046032205032049t_real @ A @ A )
= bot_bot_set_set_real ) ).
% Diff_cancel
thf(fact_675_Diff__cancel,axiom,
! [A: set_real] :
( ( minus_minus_set_real @ A @ A )
= bot_bot_set_real ) ).
% Diff_cancel
thf(fact_676_Diff__cancel,axiom,
! [A: set_nat] :
( ( minus_minus_set_nat @ A @ A )
= bot_bot_set_nat ) ).
% Diff_cancel
thf(fact_677_all__not__in__conv,axiom,
! [A: set_o] :
( ( ! [X2: $o] :
~ ( member_o @ X2 @ A ) )
= ( A = bot_bot_set_o ) ) ).
% all_not_in_conv
thf(fact_678_all__not__in__conv,axiom,
! [A: set_set_real] :
( ( ! [X2: set_real] :
~ ( member_set_real @ X2 @ A ) )
= ( A = bot_bot_set_set_real ) ) ).
% all_not_in_conv
thf(fact_679_all__not__in__conv,axiom,
! [A: set_real] :
( ( ! [X2: real] :
~ ( member_real @ X2 @ A ) )
= ( A = bot_bot_set_real ) ) ).
% all_not_in_conv
thf(fact_680_all__not__in__conv,axiom,
! [A: set_nat] :
( ( ! [X2: nat] :
~ ( member_nat @ X2 @ A ) )
= ( A = bot_bot_set_nat ) ) ).
% all_not_in_conv
thf(fact_681_Collect__empty__eq,axiom,
! [P: set_real > $o] :
( ( ( collect_set_real @ P )
= bot_bot_set_set_real )
= ( ! [X2: set_real] :
~ ( P @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_682_Collect__empty__eq,axiom,
! [P: real > $o] :
( ( ( collect_real @ P )
= bot_bot_set_real )
= ( ! [X2: real] :
~ ( P @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_683_Collect__empty__eq,axiom,
! [P: nat > $o] :
( ( ( collect_nat @ P )
= bot_bot_set_nat )
= ( ! [X2: nat] :
~ ( P @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_684_empty__Collect__eq,axiom,
! [P: set_real > $o] :
( ( bot_bot_set_set_real
= ( collect_set_real @ P ) )
= ( ! [X2: set_real] :
~ ( P @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_685_empty__Collect__eq,axiom,
! [P: real > $o] :
( ( bot_bot_set_real
= ( collect_real @ P ) )
= ( ! [X2: real] :
~ ( P @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_686_empty__Collect__eq,axiom,
! [P: nat > $o] :
( ( bot_bot_set_nat
= ( collect_nat @ P ) )
= ( ! [X2: nat] :
~ ( P @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_687_InterI,axiom,
! [C2: set_set_set_real,A: set_real] :
( ! [X5: set_set_real] :
( ( member_set_set_real @ X5 @ C2 )
=> ( member_set_real @ A @ X5 ) )
=> ( member_set_real @ A @ ( comple6920828426275262033t_real @ C2 ) ) ) ).
% InterI
thf(fact_688_InterI,axiom,
! [C2: set_set_nat,A: nat] :
( ! [X5: set_nat] :
( ( member_set_nat @ X5 @ C2 )
=> ( member_nat @ A @ X5 ) )
=> ( member_nat @ A @ ( comple7806235888213564991et_nat @ C2 ) ) ) ).
% InterI
thf(fact_689_InterI,axiom,
! [C2: set_set_o,A: $o] :
( ! [X5: set_o] :
( ( member_set_o @ X5 @ C2 )
=> ( member_o @ A @ X5 ) )
=> ( member_o @ A @ ( comple3063163877087187839_set_o @ C2 ) ) ) ).
% InterI
thf(fact_690_InterI,axiom,
! [C2: set_set_real,A: real] :
( ! [X5: set_real] :
( ( member_set_real @ X5 @ C2 )
=> ( member_real @ A @ X5 ) )
=> ( member_real @ A @ ( comple8289635161444856091t_real @ C2 ) ) ) ).
% InterI
thf(fact_691_Inter__iff,axiom,
! [A: set_real,C2: set_set_set_real] :
( ( member_set_real @ A @ ( comple6920828426275262033t_real @ C2 ) )
= ( ! [X2: set_set_real] :
( ( member_set_set_real @ X2 @ C2 )
=> ( member_set_real @ A @ X2 ) ) ) ) ).
% Inter_iff
thf(fact_692_Inter__iff,axiom,
! [A: real,C2: set_set_real] :
( ( member_real @ A @ ( comple8289635161444856091t_real @ C2 ) )
= ( ! [X2: set_real] :
( ( member_set_real @ X2 @ C2 )
=> ( member_real @ A @ X2 ) ) ) ) ).
% Inter_iff
thf(fact_693_Inter__iff,axiom,
! [A: nat,C2: set_set_nat] :
( ( member_nat @ A @ ( comple7806235888213564991et_nat @ C2 ) )
= ( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ C2 )
=> ( member_nat @ A @ X2 ) ) ) ) ).
% Inter_iff
thf(fact_694_Inter__iff,axiom,
! [A: $o,C2: set_set_o] :
( ( member_o @ A @ ( comple3063163877087187839_set_o @ C2 ) )
= ( ! [X2: set_o] :
( ( member_set_o @ X2 @ C2 )
=> ( member_o @ A @ X2 ) ) ) ) ).
% Inter_iff
thf(fact_695_image__ident,axiom,
! [Y6: set_real] :
( ( image_real_real
@ ^ [X2: real] : X2
@ Y6 )
= Y6 ) ).
% image_ident
thf(fact_696_not__gr__zero,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_697_image__empty,axiom,
! [F: set_real > set_real] :
( ( image_2436557299294012491t_real @ F @ bot_bot_set_set_real )
= bot_bot_set_set_real ) ).
% image_empty
thf(fact_698_image__empty,axiom,
! [F: set_real > real] :
( ( image_set_real_real @ F @ bot_bot_set_set_real )
= bot_bot_set_real ) ).
% image_empty
thf(fact_699_image__empty,axiom,
! [F: set_real > nat] :
( ( image_set_real_nat @ F @ bot_bot_set_set_real )
= bot_bot_set_nat ) ).
% image_empty
thf(fact_700_image__empty,axiom,
! [F: real > set_real] :
( ( image_real_set_real @ F @ bot_bot_set_real )
= bot_bot_set_set_real ) ).
% image_empty
thf(fact_701_image__empty,axiom,
! [F: real > real] :
( ( image_real_real @ F @ bot_bot_set_real )
= bot_bot_set_real ) ).
% image_empty
thf(fact_702_image__empty,axiom,
! [F: real > nat] :
( ( image_real_nat @ F @ bot_bot_set_real )
= bot_bot_set_nat ) ).
% image_empty
thf(fact_703_image__empty,axiom,
! [F: nat > int] :
( ( image_nat_int @ F @ bot_bot_set_nat )
= bot_bot_set_int ) ).
% image_empty
thf(fact_704_image__empty,axiom,
! [F: nat > set_real] :
( ( image_nat_set_real @ F @ bot_bot_set_nat )
= bot_bot_set_set_real ) ).
% image_empty
thf(fact_705_image__empty,axiom,
! [F: nat > real] :
( ( image_nat_real @ F @ bot_bot_set_nat )
= bot_bot_set_real ) ).
% image_empty
thf(fact_706_image__empty,axiom,
! [F: nat > nat] :
( ( image_nat_nat @ F @ bot_bot_set_nat )
= bot_bot_set_nat ) ).
% image_empty
thf(fact_707_empty__is__image,axiom,
! [F: nat > int,A: set_nat] :
( ( bot_bot_set_int
= ( image_nat_int @ F @ A ) )
= ( A = bot_bot_set_nat ) ) ).
% empty_is_image
thf(fact_708_empty__is__image,axiom,
! [F: set_real > set_real,A: set_set_real] :
( ( bot_bot_set_set_real
= ( image_2436557299294012491t_real @ F @ A ) )
= ( A = bot_bot_set_set_real ) ) ).
% empty_is_image
thf(fact_709_empty__is__image,axiom,
! [F: real > set_real,A: set_real] :
( ( bot_bot_set_set_real
= ( image_real_set_real @ F @ A ) )
= ( A = bot_bot_set_real ) ) ).
% empty_is_image
thf(fact_710_empty__is__image,axiom,
! [F: nat > set_real,A: set_nat] :
( ( bot_bot_set_set_real
= ( image_nat_set_real @ F @ A ) )
= ( A = bot_bot_set_nat ) ) ).
% empty_is_image
thf(fact_711_empty__is__image,axiom,
! [F: set_real > real,A: set_set_real] :
( ( bot_bot_set_real
= ( image_set_real_real @ F @ A ) )
= ( A = bot_bot_set_set_real ) ) ).
% empty_is_image
thf(fact_712_empty__is__image,axiom,
! [F: real > real,A: set_real] :
( ( bot_bot_set_real
= ( image_real_real @ F @ A ) )
= ( A = bot_bot_set_real ) ) ).
% empty_is_image
thf(fact_713_empty__is__image,axiom,
! [F: nat > real,A: set_nat] :
( ( bot_bot_set_real
= ( image_nat_real @ F @ A ) )
= ( A = bot_bot_set_nat ) ) ).
% empty_is_image
thf(fact_714_empty__is__image,axiom,
! [F: set_real > nat,A: set_set_real] :
( ( bot_bot_set_nat
= ( image_set_real_nat @ F @ A ) )
= ( A = bot_bot_set_set_real ) ) ).
% empty_is_image
thf(fact_715_empty__is__image,axiom,
! [F: real > nat,A: set_real] :
( ( bot_bot_set_nat
= ( image_real_nat @ F @ A ) )
= ( A = bot_bot_set_real ) ) ).
% empty_is_image
thf(fact_716_empty__is__image,axiom,
! [F: nat > nat,A: set_nat] :
( ( bot_bot_set_nat
= ( image_nat_nat @ F @ A ) )
= ( A = bot_bot_set_nat ) ) ).
% empty_is_image
thf(fact_717_image__is__empty,axiom,
! [F: nat > int,A: set_nat] :
( ( ( image_nat_int @ F @ A )
= bot_bot_set_int )
= ( A = bot_bot_set_nat ) ) ).
% image_is_empty
thf(fact_718_image__is__empty,axiom,
! [F: set_real > set_real,A: set_set_real] :
( ( ( image_2436557299294012491t_real @ F @ A )
= bot_bot_set_set_real )
= ( A = bot_bot_set_set_real ) ) ).
% image_is_empty
thf(fact_719_image__is__empty,axiom,
! [F: real > set_real,A: set_real] :
( ( ( image_real_set_real @ F @ A )
= bot_bot_set_set_real )
= ( A = bot_bot_set_real ) ) ).
% image_is_empty
thf(fact_720_image__is__empty,axiom,
! [F: nat > set_real,A: set_nat] :
( ( ( image_nat_set_real @ F @ A )
= bot_bot_set_set_real )
= ( A = bot_bot_set_nat ) ) ).
% image_is_empty
thf(fact_721_image__is__empty,axiom,
! [F: set_real > real,A: set_set_real] :
( ( ( image_set_real_real @ F @ A )
= bot_bot_set_real )
= ( A = bot_bot_set_set_real ) ) ).
% image_is_empty
thf(fact_722_image__is__empty,axiom,
! [F: real > real,A: set_real] :
( ( ( image_real_real @ F @ A )
= bot_bot_set_real )
= ( A = bot_bot_set_real ) ) ).
% image_is_empty
thf(fact_723_image__is__empty,axiom,
! [F: nat > real,A: set_nat] :
( ( ( image_nat_real @ F @ A )
= bot_bot_set_real )
= ( A = bot_bot_set_nat ) ) ).
% image_is_empty
thf(fact_724_image__is__empty,axiom,
! [F: set_real > nat,A: set_set_real] :
( ( ( image_set_real_nat @ F @ A )
= bot_bot_set_nat )
= ( A = bot_bot_set_set_real ) ) ).
% image_is_empty
thf(fact_725_image__is__empty,axiom,
! [F: real > nat,A: set_real] :
( ( ( image_real_nat @ F @ A )
= bot_bot_set_nat )
= ( A = bot_bot_set_real ) ) ).
% image_is_empty
thf(fact_726_image__is__empty,axiom,
! [F: nat > nat,A: set_nat] :
( ( ( image_nat_nat @ F @ A )
= bot_bot_set_nat )
= ( A = bot_bot_set_nat ) ) ).
% image_is_empty
thf(fact_727_Diff__eq__empty__iff,axiom,
! [A: set_set_real,B4: set_set_real] :
( ( ( minus_5467046032205032049t_real @ A @ B4 )
= bot_bot_set_set_real )
= ( ord_le3558479182127378552t_real @ A @ B4 ) ) ).
% Diff_eq_empty_iff
thf(fact_728_Diff__eq__empty__iff,axiom,
! [A: set_nat,B4: set_nat] :
( ( ( minus_minus_set_nat @ A @ B4 )
= bot_bot_set_nat )
= ( ord_less_eq_set_nat @ A @ B4 ) ) ).
% Diff_eq_empty_iff
thf(fact_729_Diff__eq__empty__iff,axiom,
! [A: set_real,B4: set_real] :
( ( ( minus_minus_set_real @ A @ B4 )
= bot_bot_set_real )
= ( ord_less_eq_set_real @ A @ B4 ) ) ).
% Diff_eq_empty_iff
thf(fact_730_empty__subsetI,axiom,
! [A: set_set_real] : ( ord_le3558479182127378552t_real @ bot_bot_set_set_real @ A ) ).
% empty_subsetI
thf(fact_731_empty__subsetI,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A ) ).
% empty_subsetI
thf(fact_732_empty__subsetI,axiom,
! [A: set_real] : ( ord_less_eq_set_real @ bot_bot_set_real @ A ) ).
% empty_subsetI
thf(fact_733_of__nat__less__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_iff
thf(fact_734_of__nat__less__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_iff
thf(fact_735_of__nat__less__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_iff
thf(fact_736_SUP__identity__eq,axiom,
! [A: set_real] :
( ( comple1385675409528146559p_real
@ ( image_real_real
@ ^ [X2: real] : X2
@ A ) )
= ( comple1385675409528146559p_real @ A ) ) ).
% SUP_identity_eq
thf(fact_737_SUP__identity__eq,axiom,
! [A: set_set_real] :
( ( comple3096694443085538997t_real
@ ( image_2436557299294012491t_real
@ ^ [X2: set_real] : X2
@ A ) )
= ( comple3096694443085538997t_real @ A ) ) ).
% SUP_identity_eq
thf(fact_738_SUP__identity__eq,axiom,
! [A: set_nat] :
( ( complete_Sup_Sup_nat
@ ( image_nat_nat
@ ^ [X2: nat] : X2
@ A ) )
= ( complete_Sup_Sup_nat @ A ) ) ).
% SUP_identity_eq
thf(fact_739_SUP__identity__eq,axiom,
! [A: set_o] :
( ( complete_Sup_Sup_o
@ ( image_o_o
@ ^ [X2: $o] : X2
@ A ) )
= ( complete_Sup_Sup_o @ A ) ) ).
% SUP_identity_eq
thf(fact_740_INF__identity__eq,axiom,
! [A: set_real] :
( ( comple4887499456419720421f_real
@ ( image_real_real
@ ^ [X2: real] : X2
@ A ) )
= ( comple4887499456419720421f_real @ A ) ) ).
% INF_identity_eq
thf(fact_741_INF__identity__eq,axiom,
! [A: set_nat] :
( ( complete_Inf_Inf_nat
@ ( image_nat_nat
@ ^ [X2: nat] : X2
@ A ) )
= ( complete_Inf_Inf_nat @ A ) ) ).
% INF_identity_eq
thf(fact_742_INF__identity__eq,axiom,
! [A: set_o] :
( ( complete_Inf_Inf_o
@ ( image_o_o
@ ^ [X2: $o] : X2
@ A ) )
= ( complete_Inf_Inf_o @ A ) ) ).
% INF_identity_eq
thf(fact_743_diff__gt__0__iff__gt,axiom,
! [A2: real,B: real] :
( ( ord_less_real @ zero_zero_real @ ( minus_minus_real @ A2 @ B ) )
= ( ord_less_real @ B @ A2 ) ) ).
% diff_gt_0_iff_gt
thf(fact_744_diff__gt__0__iff__gt,axiom,
! [A2: int,B: int] :
( ( ord_less_int @ zero_zero_int @ ( minus_minus_int @ A2 @ B ) )
= ( ord_less_int @ B @ A2 ) ) ).
% diff_gt_0_iff_gt
thf(fact_745_zero__less__abs__iff,axiom,
! [A2: real] :
( ( ord_less_real @ zero_zero_real @ ( abs_abs_real @ A2 ) )
= ( A2 != zero_zero_real ) ) ).
% zero_less_abs_iff
thf(fact_746_zero__less__abs__iff,axiom,
! [A2: int] :
( ( ord_less_int @ zero_zero_int @ ( abs_abs_int @ A2 ) )
= ( A2 != zero_zero_int ) ) ).
% zero_less_abs_iff
thf(fact_747_atLeastatMost__empty,axiom,
! [B: set_real,A2: set_real] :
( ( ord_less_set_real @ B @ A2 )
=> ( ( set_or7743017856606604397t_real @ A2 @ B )
= bot_bot_set_set_real ) ) ).
% atLeastatMost_empty
thf(fact_748_atLeastatMost__empty,axiom,
! [B: real,A2: real] :
( ( ord_less_real @ B @ A2 )
=> ( ( set_or1222579329274155063t_real @ A2 @ B )
= bot_bot_set_real ) ) ).
% atLeastatMost_empty
thf(fact_749_atLeastatMost__empty,axiom,
! [B: nat,A2: nat] :
( ( ord_less_nat @ B @ A2 )
=> ( ( set_or1269000886237332187st_nat @ A2 @ B )
= bot_bot_set_nat ) ) ).
% atLeastatMost_empty
thf(fact_750_atLeastatMost__empty,axiom,
! [B: int,A2: int] :
( ( ord_less_int @ B @ A2 )
=> ( ( set_or1266510415728281911st_int @ A2 @ B )
= bot_bot_set_int ) ) ).
% atLeastatMost_empty
thf(fact_751_image__diff__atLeastAtMost,axiom,
! [D: real,A2: real,B: real] :
( ( image_real_real @ ( minus_minus_real @ D ) @ ( set_or1222579329274155063t_real @ A2 @ B ) )
= ( set_or1222579329274155063t_real @ ( minus_minus_real @ D @ B ) @ ( minus_minus_real @ D @ A2 ) ) ) ).
% image_diff_atLeastAtMost
thf(fact_752_image__diff__atLeastAtMost,axiom,
! [D: int,A2: int,B: int] :
( ( image_int_int @ ( minus_minus_int @ D ) @ ( set_or1266510415728281911st_int @ A2 @ B ) )
= ( set_or1266510415728281911st_int @ ( minus_minus_int @ D @ B ) @ ( minus_minus_int @ D @ A2 ) ) ) ).
% image_diff_atLeastAtMost
thf(fact_753_not__real__square__gt__zero,axiom,
! [X: real] :
( ( ~ ( ord_less_real @ zero_zero_real @ ( times_times_real @ X @ X ) ) )
= ( X = zero_zero_real ) ) ).
% not_real_square_gt_zero
thf(fact_754_a__seg__less__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ ( a_seg @ X ) @ ( a_seg @ Y ) )
= ( ord_less_real @ X @ Y ) ) ).
% a_seg_less_iff
thf(fact_755_ccSUP__const,axiom,
! [A: set_set_real,F: set_real] :
( ( A != bot_bot_set_set_real )
=> ( ( comple3096694443085538997t_real
@ ( image_2436557299294012491t_real
@ ^ [I: set_real] : F
@ A ) )
= F ) ) ).
% ccSUP_const
thf(fact_756_ccSUP__const,axiom,
! [A: set_real,F: set_real] :
( ( A != bot_bot_set_real )
=> ( ( comple3096694443085538997t_real
@ ( image_real_set_real
@ ^ [I: real] : F
@ A ) )
= F ) ) ).
% ccSUP_const
thf(fact_757_ccSUP__const,axiom,
! [A: set_nat,F: set_real] :
( ( A != bot_bot_set_nat )
=> ( ( comple3096694443085538997t_real
@ ( image_nat_set_real
@ ^ [I: nat] : F
@ A ) )
= F ) ) ).
% ccSUP_const
thf(fact_758_ccSUP__const,axiom,
! [A: set_set_real,F: $o] :
( ( A != bot_bot_set_set_real )
=> ( ( complete_Sup_Sup_o
@ ( image_set_real_o
@ ^ [I: set_real] : F
@ A ) )
= F ) ) ).
% ccSUP_const
thf(fact_759_ccSUP__const,axiom,
! [A: set_real,F: $o] :
( ( A != bot_bot_set_real )
=> ( ( complete_Sup_Sup_o
@ ( image_real_o
@ ^ [I: real] : F
@ A ) )
= F ) ) ).
% ccSUP_const
thf(fact_760_ccSUP__const,axiom,
! [A: set_nat,F: $o] :
( ( A != bot_bot_set_nat )
=> ( ( complete_Sup_Sup_o
@ ( image_nat_o
@ ^ [I: nat] : F
@ A ) )
= F ) ) ).
% ccSUP_const
thf(fact_761_SUP__const,axiom,
! [A: set_set_real,F: set_real] :
( ( A != bot_bot_set_set_real )
=> ( ( comple3096694443085538997t_real
@ ( image_2436557299294012491t_real
@ ^ [I: set_real] : F
@ A ) )
= F ) ) ).
% SUP_const
thf(fact_762_SUP__const,axiom,
! [A: set_real,F: set_real] :
( ( A != bot_bot_set_real )
=> ( ( comple3096694443085538997t_real
@ ( image_real_set_real
@ ^ [I: real] : F
@ A ) )
= F ) ) ).
% SUP_const
thf(fact_763_SUP__const,axiom,
! [A: set_nat,F: set_real] :
( ( A != bot_bot_set_nat )
=> ( ( comple3096694443085538997t_real
@ ( image_nat_set_real
@ ^ [I: nat] : F
@ A ) )
= F ) ) ).
% SUP_const
thf(fact_764_SUP__const,axiom,
! [A: set_set_real,F: $o] :
( ( A != bot_bot_set_set_real )
=> ( ( complete_Sup_Sup_o
@ ( image_set_real_o
@ ^ [I: set_real] : F
@ A ) )
= F ) ) ).
% SUP_const
thf(fact_765_SUP__const,axiom,
! [A: set_real,F: $o] :
( ( A != bot_bot_set_real )
=> ( ( complete_Sup_Sup_o
@ ( image_real_o
@ ^ [I: real] : F
@ A ) )
= F ) ) ).
% SUP_const
thf(fact_766_SUP__const,axiom,
! [A: set_nat,F: $o] :
( ( A != bot_bot_set_nat )
=> ( ( complete_Sup_Sup_o
@ ( image_nat_o
@ ^ [I: nat] : F
@ A ) )
= F ) ) ).
% SUP_const
thf(fact_767_cSUP__const,axiom,
! [A: set_nat,C: int] :
( ( A != bot_bot_set_nat )
=> ( ( complete_Sup_Sup_int
@ ( image_nat_int
@ ^ [X2: nat] : C
@ A ) )
= C ) ) ).
% cSUP_const
thf(fact_768_cSUP__const,axiom,
! [A: set_real,C: real] :
( ( A != bot_bot_set_real )
=> ( ( comple1385675409528146559p_real
@ ( image_real_real
@ ^ [X2: real] : C
@ A ) )
= C ) ) ).
% cSUP_const
thf(fact_769_cSUP__const,axiom,
! [A: set_nat,C: real] :
( ( A != bot_bot_set_nat )
=> ( ( comple1385675409528146559p_real
@ ( image_nat_real
@ ^ [X2: nat] : C
@ A ) )
= C ) ) ).
% cSUP_const
thf(fact_770_cSUP__const,axiom,
! [A: set_real,C: nat] :
( ( A != bot_bot_set_real )
=> ( ( complete_Sup_Sup_nat
@ ( image_real_nat
@ ^ [X2: real] : C
@ A ) )
= C ) ) ).
% cSUP_const
thf(fact_771_cSUP__const,axiom,
! [A: set_nat,C: nat] :
( ( A != bot_bot_set_nat )
=> ( ( complete_Sup_Sup_nat
@ ( image_nat_nat
@ ^ [X2: nat] : C
@ A ) )
= C ) ) ).
% cSUP_const
thf(fact_772_cSUP__const,axiom,
! [A: set_real,C: $o] :
( ( A != bot_bot_set_real )
=> ( ( complete_Sup_Sup_o
@ ( image_real_o
@ ^ [X2: real] : C
@ A ) )
= C ) ) ).
% cSUP_const
thf(fact_773_cSUP__const,axiom,
! [A: set_nat,C: $o] :
( ( A != bot_bot_set_nat )
=> ( ( complete_Sup_Sup_o
@ ( image_nat_o
@ ^ [X2: nat] : C
@ A ) )
= C ) ) ).
% cSUP_const
thf(fact_774_cSUP__const,axiom,
! [A: set_set_real,C: real] :
( ( A != bot_bot_set_set_real )
=> ( ( comple1385675409528146559p_real
@ ( image_set_real_real
@ ^ [X2: set_real] : C
@ A ) )
= C ) ) ).
% cSUP_const
thf(fact_775_cSUP__const,axiom,
! [A: set_real,C: set_real] :
( ( A != bot_bot_set_real )
=> ( ( comple3096694443085538997t_real
@ ( image_real_set_real
@ ^ [X2: real] : C
@ A ) )
= C ) ) ).
% cSUP_const
thf(fact_776_cSUP__const,axiom,
! [A: set_nat,C: set_real] :
( ( A != bot_bot_set_nat )
=> ( ( comple3096694443085538997t_real
@ ( image_nat_set_real
@ ^ [X2: nat] : C
@ A ) )
= C ) ) ).
% cSUP_const
thf(fact_777_image__minus__const__atLeastAtMost_H,axiom,
! [D: real,A2: real,B: real] :
( ( image_real_real
@ ^ [T3: real] : ( minus_minus_real @ T3 @ D )
@ ( set_or1222579329274155063t_real @ A2 @ B ) )
= ( set_or1222579329274155063t_real @ ( minus_minus_real @ A2 @ D ) @ ( minus_minus_real @ B @ D ) ) ) ).
% image_minus_const_atLeastAtMost'
thf(fact_778_image__minus__const__atLeastAtMost_H,axiom,
! [D: int,A2: int,B: int] :
( ( image_int_int
@ ^ [T3: int] : ( minus_minus_int @ T3 @ D )
@ ( set_or1266510415728281911st_int @ A2 @ B ) )
= ( set_or1266510415728281911st_int @ ( minus_minus_int @ A2 @ D ) @ ( minus_minus_int @ B @ D ) ) ) ).
% image_minus_const_atLeastAtMost'
thf(fact_779_ccINF__const,axiom,
! [A: set_set_real,F: $o] :
( ( A != bot_bot_set_set_real )
=> ( ( complete_Inf_Inf_o
@ ( image_set_real_o
@ ^ [I: set_real] : F
@ A ) )
= F ) ) ).
% ccINF_const
thf(fact_780_ccINF__const,axiom,
! [A: set_real,F: $o] :
( ( A != bot_bot_set_real )
=> ( ( complete_Inf_Inf_o
@ ( image_real_o
@ ^ [I: real] : F
@ A ) )
= F ) ) ).
% ccINF_const
thf(fact_781_ccINF__const,axiom,
! [A: set_nat,F: $o] :
( ( A != bot_bot_set_nat )
=> ( ( complete_Inf_Inf_o
@ ( image_nat_o
@ ^ [I: nat] : F
@ A ) )
= F ) ) ).
% ccINF_const
thf(fact_782_INF__const,axiom,
! [A: set_set_real,F: $o] :
( ( A != bot_bot_set_set_real )
=> ( ( complete_Inf_Inf_o
@ ( image_set_real_o
@ ^ [I: set_real] : F
@ A ) )
= F ) ) ).
% INF_const
thf(fact_783_INF__const,axiom,
! [A: set_real,F: $o] :
( ( A != bot_bot_set_real )
=> ( ( complete_Inf_Inf_o
@ ( image_real_o
@ ^ [I: real] : F
@ A ) )
= F ) ) ).
% INF_const
thf(fact_784_INF__const,axiom,
! [A: set_nat,F: $o] :
( ( A != bot_bot_set_nat )
=> ( ( complete_Inf_Inf_o
@ ( image_nat_o
@ ^ [I: nat] : F
@ A ) )
= F ) ) ).
% INF_const
thf(fact_785_cINF__const,axiom,
! [A: set_nat,C: int] :
( ( A != bot_bot_set_nat )
=> ( ( complete_Inf_Inf_int
@ ( image_nat_int
@ ^ [X2: nat] : C
@ A ) )
= C ) ) ).
% cINF_const
thf(fact_786_cINF__const,axiom,
! [A: set_set_real,C: real] :
( ( A != bot_bot_set_set_real )
=> ( ( comple4887499456419720421f_real
@ ( image_set_real_real
@ ^ [X2: set_real] : C
@ A ) )
= C ) ) ).
% cINF_const
thf(fact_787_cINF__const,axiom,
! [A: set_real,C: real] :
( ( A != bot_bot_set_real )
=> ( ( comple4887499456419720421f_real
@ ( image_real_real
@ ^ [X2: real] : C
@ A ) )
= C ) ) ).
% cINF_const
thf(fact_788_cINF__const,axiom,
! [A: set_nat,C: real] :
( ( A != bot_bot_set_nat )
=> ( ( comple4887499456419720421f_real
@ ( image_nat_real
@ ^ [X2: nat] : C
@ A ) )
= C ) ) ).
% cINF_const
thf(fact_789_cINF__const,axiom,
! [A: set_set_real,C: nat] :
( ( A != bot_bot_set_set_real )
=> ( ( complete_Inf_Inf_nat
@ ( image_set_real_nat
@ ^ [X2: set_real] : C
@ A ) )
= C ) ) ).
% cINF_const
thf(fact_790_cINF__const,axiom,
! [A: set_real,C: nat] :
( ( A != bot_bot_set_real )
=> ( ( complete_Inf_Inf_nat
@ ( image_real_nat
@ ^ [X2: real] : C
@ A ) )
= C ) ) ).
% cINF_const
thf(fact_791_cINF__const,axiom,
! [A: set_nat,C: nat] :
( ( A != bot_bot_set_nat )
=> ( ( complete_Inf_Inf_nat
@ ( image_nat_nat
@ ^ [X2: nat] : C
@ A ) )
= C ) ) ).
% cINF_const
thf(fact_792_cINF__const,axiom,
! [A: set_set_real,C: $o] :
( ( A != bot_bot_set_set_real )
=> ( ( complete_Inf_Inf_o
@ ( image_set_real_o
@ ^ [X2: set_real] : C
@ A ) )
= C ) ) ).
% cINF_const
thf(fact_793_cINF__const,axiom,
! [A: set_real,C: $o] :
( ( A != bot_bot_set_real )
=> ( ( complete_Inf_Inf_o
@ ( image_real_o
@ ^ [X2: real] : C
@ A ) )
= C ) ) ).
% cINF_const
thf(fact_794_cINF__const,axiom,
! [A: set_nat,C: $o] :
( ( A != bot_bot_set_nat )
=> ( ( complete_Inf_Inf_o
@ ( image_nat_o
@ ^ [X2: nat] : C
@ A ) )
= C ) ) ).
% cINF_const
thf(fact_795_of__nat__0__less__iff,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N ) )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% of_nat_0_less_iff
thf(fact_796_of__nat__0__less__iff,axiom,
! [N: nat] :
( ( ord_less_real @ zero_zero_real @ ( semiri5074537144036343181t_real @ N ) )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% of_nat_0_less_iff
thf(fact_797_of__nat__0__less__iff,axiom,
! [N: nat] :
( ( ord_less_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% of_nat_0_less_iff
thf(fact_798_ccSUP__empty,axiom,
! [F: real > $o] :
( ( complete_Sup_Sup_o @ ( image_real_o @ F @ bot_bot_set_real ) )
= bot_bot_o ) ).
% ccSUP_empty
thf(fact_799_ccSUP__empty,axiom,
! [F: nat > $o] :
( ( complete_Sup_Sup_o @ ( image_nat_o @ F @ bot_bot_set_nat ) )
= bot_bot_o ) ).
% ccSUP_empty
thf(fact_800_ccSUP__empty,axiom,
! [F: real > set_nat] :
( ( comple7399068483239264473et_nat @ ( image_real_set_nat @ F @ bot_bot_set_real ) )
= bot_bot_set_nat ) ).
% ccSUP_empty
thf(fact_801_ccSUP__empty,axiom,
! [F: nat > set_nat] :
( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ bot_bot_set_nat ) )
= bot_bot_set_nat ) ).
% ccSUP_empty
thf(fact_802_ccSUP__empty,axiom,
! [F: real > set_real] :
( ( comple3096694443085538997t_real @ ( image_real_set_real @ F @ bot_bot_set_real ) )
= bot_bot_set_real ) ).
% ccSUP_empty
thf(fact_803_ccSUP__empty,axiom,
! [F: nat > set_real] :
( ( comple3096694443085538997t_real @ ( image_nat_set_real @ F @ bot_bot_set_nat ) )
= bot_bot_set_real ) ).
% ccSUP_empty
thf(fact_804_ccSUP__empty,axiom,
! [F: set_real > $o] :
( ( complete_Sup_Sup_o @ ( image_set_real_o @ F @ bot_bot_set_set_real ) )
= bot_bot_o ) ).
% ccSUP_empty
thf(fact_805_ccSUP__empty,axiom,
! [F: set_real > set_nat] :
( ( comple7399068483239264473et_nat @ ( image_7270232309134952815et_nat @ F @ bot_bot_set_set_real ) )
= bot_bot_set_nat ) ).
% ccSUP_empty
thf(fact_806_ccSUP__empty,axiom,
! [F: real > set_set_real] :
( ( comple5917660045593844715t_real @ ( image_3243600997494576203t_real @ F @ bot_bot_set_real ) )
= bot_bot_set_set_real ) ).
% ccSUP_empty
thf(fact_807_ccSUP__empty,axiom,
! [F: nat > set_set_real] :
( ( comple5917660045593844715t_real @ ( image_396256051147326063t_real @ F @ bot_bot_set_nat ) )
= bot_bot_set_set_real ) ).
% ccSUP_empty
thf(fact_808_f__iff_I1_J,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ( ord_less_real @ ( f @ X ) @ ( f @ Y ) )
= ( ord_less_real @ X @ Y ) ) ) ) ).
% f_iff(1)
thf(fact_809_image__mult__atLeastAtMost,axiom,
! [D: real,A2: real,B: real] :
( ( ord_less_real @ zero_zero_real @ D )
=> ( ( image_real_real @ ( times_times_real @ D ) @ ( set_or1222579329274155063t_real @ A2 @ B ) )
= ( set_or1222579329274155063t_real @ ( times_times_real @ D @ A2 ) @ ( times_times_real @ D @ B ) ) ) ) ).
% image_mult_atLeastAtMost
thf(fact_810_image__divide__atLeastAtMost,axiom,
! [D: real,A2: real,B: real] :
( ( ord_less_real @ zero_zero_real @ D )
=> ( ( image_real_real
@ ^ [C3: real] : ( divide_divide_real @ C3 @ D )
@ ( set_or1222579329274155063t_real @ A2 @ B ) )
= ( set_or1222579329274155063t_real @ ( divide_divide_real @ A2 @ D ) @ ( divide_divide_real @ B @ D ) ) ) ) ).
% image_divide_atLeastAtMost
thf(fact_811_del__gt0,axiom,
! [E2: real] :
( ( ord_less_real @ zero_zero_real @ E2 )
=> ( ord_less_real @ zero_zero_real @ ( del @ E2 ) ) ) ).
% del_gt0
thf(fact_812_less,axiom,
! [K2: set_real] :
( ( member_set_real @ K2 @ ( regular_division @ zero_zero_real @ a @ n ) )
=> ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ ( f @ ( comple1385675409528146559p_real @ K2 ) ) @ ( f @ ( comple4887499456419720421f_real @ K2 ) ) ) ) @ ( divide_divide_real @ epsilon @ a ) ) ) ).
% less
thf(fact_813_del,axiom,
! [X8: real,X: real,E2: real] :
( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X8 @ X ) ) @ ( del @ E2 ) )
=> ( ( ord_less_real @ zero_zero_real @ E2 )
=> ( ( member_real @ X @ ( set_or1222579329274155063t_real @ zero_zero_real @ a ) )
=> ( ( member_real @ X8 @ ( set_or1222579329274155063t_real @ zero_zero_real @ a ) )
=> ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ ( f @ X8 ) @ ( f @ X ) ) ) @ E2 ) ) ) ) ) ).
% del
thf(fact_814_image__diff__subset,axiom,
! [F: nat > int,A: set_nat,B4: set_nat] : ( ord_less_eq_set_int @ ( minus_minus_set_int @ ( image_nat_int @ F @ A ) @ ( image_nat_int @ F @ B4 ) ) @ ( image_nat_int @ F @ ( minus_minus_set_nat @ A @ B4 ) ) ) ).
% image_diff_subset
thf(fact_815_image__diff__subset,axiom,
! [F: real > real,A: set_real,B4: set_real] : ( ord_less_eq_set_real @ ( minus_minus_set_real @ ( image_real_real @ F @ A ) @ ( image_real_real @ F @ B4 ) ) @ ( image_real_real @ F @ ( minus_minus_set_real @ A @ B4 ) ) ) ).
% image_diff_subset
thf(fact_816_Inf__nat__def1,axiom,
! [K2: set_nat] :
( ( K2 != bot_bot_set_nat )
=> ( member_nat @ ( complete_Inf_Inf_nat @ K2 ) @ K2 ) ) ).
% Inf_nat_def1
thf(fact_817_less__INF__D,axiom,
! [Y: $o,F: set_real > $o,A: set_set_real,I2: set_real] :
( ( ord_less_o @ Y @ ( complete_Inf_Inf_o @ ( image_set_real_o @ F @ A ) ) )
=> ( ( member_set_real @ I2 @ A )
=> ( ord_less_o @ Y @ ( F @ I2 ) ) ) ) ).
% less_INF_D
thf(fact_818_less__INF__D,axiom,
! [Y: $o,F: real > $o,A: set_real,I2: real] :
( ( ord_less_o @ Y @ ( complete_Inf_Inf_o @ ( image_real_o @ F @ A ) ) )
=> ( ( member_real @ I2 @ A )
=> ( ord_less_o @ Y @ ( F @ I2 ) ) ) ) ).
% less_INF_D
thf(fact_819_less__INF__D,axiom,
! [Y: $o,F: nat > $o,A: set_nat,I2: nat] :
( ( ord_less_o @ Y @ ( complete_Inf_Inf_o @ ( image_nat_o @ F @ A ) ) )
=> ( ( member_nat @ I2 @ A )
=> ( ord_less_o @ Y @ ( F @ I2 ) ) ) ) ).
% less_INF_D
thf(fact_820_less__INF__D,axiom,
! [Y: $o,F: $o > $o,A: set_o,I2: $o] :
( ( ord_less_o @ Y @ ( complete_Inf_Inf_o @ ( image_o_o @ F @ A ) ) )
=> ( ( member_o @ I2 @ A )
=> ( ord_less_o @ Y @ ( F @ I2 ) ) ) ) ).
% less_INF_D
thf(fact_821_UN__extend__simps_I10_J,axiom,
! [B4: real > set_real,F: real > real,A: set_real] :
( ( comple3096694443085538997t_real
@ ( image_real_set_real
@ ^ [A3: real] : ( B4 @ ( F @ A3 ) )
@ A ) )
= ( comple3096694443085538997t_real @ ( image_real_set_real @ B4 @ ( image_real_real @ F @ A ) ) ) ) ).
% UN_extend_simps(10)
thf(fact_822_UN__extend__simps_I10_J,axiom,
! [B4: int > set_real,F: nat > int,A: set_nat] :
( ( comple3096694443085538997t_real
@ ( image_nat_set_real
@ ^ [A3: nat] : ( B4 @ ( F @ A3 ) )
@ A ) )
= ( comple3096694443085538997t_real @ ( image_int_set_real @ B4 @ ( image_nat_int @ F @ A ) ) ) ) ).
% UN_extend_simps(10)
thf(fact_823_InterD,axiom,
! [A: set_real,C2: set_set_set_real,X6: set_set_real] :
( ( member_set_real @ A @ ( comple6920828426275262033t_real @ C2 ) )
=> ( ( member_set_set_real @ X6 @ C2 )
=> ( member_set_real @ A @ X6 ) ) ) ).
% InterD
thf(fact_824_InterD,axiom,
! [A: real,C2: set_set_real,X6: set_real] :
( ( member_real @ A @ ( comple8289635161444856091t_real @ C2 ) )
=> ( ( member_set_real @ X6 @ C2 )
=> ( member_real @ A @ X6 ) ) ) ).
% InterD
thf(fact_825_InterD,axiom,
! [A: nat,C2: set_set_nat,X6: set_nat] :
( ( member_nat @ A @ ( comple7806235888213564991et_nat @ C2 ) )
=> ( ( member_set_nat @ X6 @ C2 )
=> ( member_nat @ A @ X6 ) ) ) ).
% InterD
thf(fact_826_InterD,axiom,
! [A: $o,C2: set_set_o,X6: set_o] :
( ( member_o @ A @ ( comple3063163877087187839_set_o @ C2 ) )
=> ( ( member_set_o @ X6 @ C2 )
=> ( member_o @ A @ X6 ) ) ) ).
% InterD
thf(fact_827_InterE,axiom,
! [A: set_real,C2: set_set_set_real,X6: set_set_real] :
( ( member_set_real @ A @ ( comple6920828426275262033t_real @ C2 ) )
=> ( ( member_set_set_real @ X6 @ C2 )
=> ( member_set_real @ A @ X6 ) ) ) ).
% InterE
thf(fact_828_InterE,axiom,
! [A: real,C2: set_set_real,X6: set_real] :
( ( member_real @ A @ ( comple8289635161444856091t_real @ C2 ) )
=> ( ( member_set_real @ X6 @ C2 )
=> ( member_real @ A @ X6 ) ) ) ).
% InterE
thf(fact_829_InterE,axiom,
! [A: nat,C2: set_set_nat,X6: set_nat] :
( ( member_nat @ A @ ( comple7806235888213564991et_nat @ C2 ) )
=> ( ( member_set_nat @ X6 @ C2 )
=> ( member_nat @ A @ X6 ) ) ) ).
% InterE
thf(fact_830_InterE,axiom,
! [A: $o,C2: set_set_o,X6: set_o] :
( ( member_o @ A @ ( comple3063163877087187839_set_o @ C2 ) )
=> ( ( member_set_o @ X6 @ C2 )
=> ( member_o @ A @ X6 ) ) ) ).
% InterE
thf(fact_831_image__Union,axiom,
! [F: nat > int,S: set_set_nat] :
( ( image_nat_int @ F @ ( comple7399068483239264473et_nat @ S ) )
= ( comple3221217463730067765et_int @ ( image_3739036796817536367et_int @ ( image_nat_int @ F ) @ S ) ) ) ).
% image_Union
thf(fact_832_image__Union,axiom,
! [F: real > real,S: set_set_real] :
( ( image_real_real @ F @ ( comple3096694443085538997t_real @ S ) )
= ( comple3096694443085538997t_real @ ( image_2436557299294012491t_real @ ( image_real_real @ F ) @ S ) ) ) ).
% image_Union
thf(fact_833_SUP__lessD,axiom,
! [F: set_real > set_real,A: set_set_real,Y: set_real,I2: set_real] :
( ( ord_less_set_real @ ( comple3096694443085538997t_real @ ( image_2436557299294012491t_real @ F @ A ) ) @ Y )
=> ( ( member_set_real @ I2 @ A )
=> ( ord_less_set_real @ ( F @ I2 ) @ Y ) ) ) ).
% SUP_lessD
thf(fact_834_SUP__lessD,axiom,
! [F: real > set_real,A: set_real,Y: set_real,I2: real] :
( ( ord_less_set_real @ ( comple3096694443085538997t_real @ ( image_real_set_real @ F @ A ) ) @ Y )
=> ( ( member_real @ I2 @ A )
=> ( ord_less_set_real @ ( F @ I2 ) @ Y ) ) ) ).
% SUP_lessD
thf(fact_835_SUP__lessD,axiom,
! [F: nat > set_real,A: set_nat,Y: set_real,I2: nat] :
( ( ord_less_set_real @ ( comple3096694443085538997t_real @ ( image_nat_set_real @ F @ A ) ) @ Y )
=> ( ( member_nat @ I2 @ A )
=> ( ord_less_set_real @ ( F @ I2 ) @ Y ) ) ) ).
% SUP_lessD
thf(fact_836_SUP__lessD,axiom,
! [F: $o > set_real,A: set_o,Y: set_real,I2: $o] :
( ( ord_less_set_real @ ( comple3096694443085538997t_real @ ( image_o_set_real @ F @ A ) ) @ Y )
=> ( ( member_o @ I2 @ A )
=> ( ord_less_set_real @ ( F @ I2 ) @ Y ) ) ) ).
% SUP_lessD
thf(fact_837_SUP__lessD,axiom,
! [F: set_real > $o,A: set_set_real,Y: $o,I2: set_real] :
( ( ord_less_o @ ( complete_Sup_Sup_o @ ( image_set_real_o @ F @ A ) ) @ Y )
=> ( ( member_set_real @ I2 @ A )
=> ( ord_less_o @ ( F @ I2 ) @ Y ) ) ) ).
% SUP_lessD
thf(fact_838_SUP__lessD,axiom,
! [F: real > $o,A: set_real,Y: $o,I2: real] :
( ( ord_less_o @ ( complete_Sup_Sup_o @ ( image_real_o @ F @ A ) ) @ Y )
=> ( ( member_real @ I2 @ A )
=> ( ord_less_o @ ( F @ I2 ) @ Y ) ) ) ).
% SUP_lessD
thf(fact_839_SUP__lessD,axiom,
! [F: nat > $o,A: set_nat,Y: $o,I2: nat] :
( ( ord_less_o @ ( complete_Sup_Sup_o @ ( image_nat_o @ F @ A ) ) @ Y )
=> ( ( member_nat @ I2 @ A )
=> ( ord_less_o @ ( F @ I2 ) @ Y ) ) ) ).
% SUP_lessD
thf(fact_840_SUP__lessD,axiom,
! [F: $o > $o,A: set_o,Y: $o,I2: $o] :
( ( ord_less_o @ ( complete_Sup_Sup_o @ ( image_o_o @ F @ A ) ) @ Y )
=> ( ( member_o @ I2 @ A )
=> ( ord_less_o @ ( F @ I2 ) @ Y ) ) ) ).
% SUP_lessD
thf(fact_841_subset__image__iff,axiom,
! [B4: set_int,F: nat > int,A: set_nat] :
( ( ord_less_eq_set_int @ B4 @ ( image_nat_int @ F @ A ) )
= ( ? [AA: set_nat] :
( ( ord_less_eq_set_nat @ AA @ A )
& ( B4
= ( image_nat_int @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_842_subset__image__iff,axiom,
! [B4: set_real,F: real > real,A: set_real] :
( ( ord_less_eq_set_real @ B4 @ ( image_real_real @ F @ A ) )
= ( ? [AA: set_real] :
( ( ord_less_eq_set_real @ AA @ A )
& ( B4
= ( image_real_real @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_843_image__subset__iff,axiom,
! [F: nat > int,A: set_nat,B4: set_int] :
( ( ord_less_eq_set_int @ ( image_nat_int @ F @ A ) @ B4 )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( member_int @ ( F @ X2 ) @ B4 ) ) ) ) ).
% image_subset_iff
thf(fact_844_image__subset__iff,axiom,
! [F: real > real,A: set_real,B4: set_real] :
( ( ord_less_eq_set_real @ ( image_real_real @ F @ A ) @ B4 )
= ( ! [X2: real] :
( ( member_real @ X2 @ A )
=> ( member_real @ ( F @ X2 ) @ B4 ) ) ) ) ).
% image_subset_iff
thf(fact_845_subset__imageE,axiom,
! [B4: set_int,F: nat > int,A: set_nat] :
( ( ord_less_eq_set_int @ B4 @ ( image_nat_int @ F @ A ) )
=> ~ ! [C4: set_nat] :
( ( ord_less_eq_set_nat @ C4 @ A )
=> ( B4
!= ( image_nat_int @ F @ C4 ) ) ) ) ).
% subset_imageE
thf(fact_846_subset__imageE,axiom,
! [B4: set_real,F: real > real,A: set_real] :
( ( ord_less_eq_set_real @ B4 @ ( image_real_real @ F @ A ) )
=> ~ ! [C4: set_real] :
( ( ord_less_eq_set_real @ C4 @ A )
=> ( B4
!= ( image_real_real @ F @ C4 ) ) ) ) ).
% subset_imageE
thf(fact_847_image__subsetI,axiom,
! [A: set_real,F: real > nat,B4: set_nat] :
( ! [X3: real] :
( ( member_real @ X3 @ A )
=> ( member_nat @ ( F @ X3 ) @ B4 ) )
=> ( ord_less_eq_set_nat @ ( image_real_nat @ F @ A ) @ B4 ) ) ).
% image_subsetI
thf(fact_848_image__subsetI,axiom,
! [A: set_real,F: real > $o,B4: set_o] :
( ! [X3: real] :
( ( member_real @ X3 @ A )
=> ( member_o @ ( F @ X3 ) @ B4 ) )
=> ( ord_less_eq_set_o @ ( image_real_o @ F @ A ) @ B4 ) ) ).
% image_subsetI
thf(fact_849_image__subsetI,axiom,
! [A: set_nat,F: nat > int,B4: set_int] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( member_int @ ( F @ X3 ) @ B4 ) )
=> ( ord_less_eq_set_int @ ( image_nat_int @ F @ A ) @ B4 ) ) ).
% image_subsetI
thf(fact_850_image__subsetI,axiom,
! [A: set_nat,F: nat > nat,B4: set_nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( member_nat @ ( F @ X3 ) @ B4 ) )
=> ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A ) @ B4 ) ) ).
% image_subsetI
thf(fact_851_image__subsetI,axiom,
! [A: set_nat,F: nat > $o,B4: set_o] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( member_o @ ( F @ X3 ) @ B4 ) )
=> ( ord_less_eq_set_o @ ( image_nat_o @ F @ A ) @ B4 ) ) ).
% image_subsetI
thf(fact_852_image__subsetI,axiom,
! [A: set_o,F: $o > nat,B4: set_nat] :
( ! [X3: $o] :
( ( member_o @ X3 @ A )
=> ( member_nat @ ( F @ X3 ) @ B4 ) )
=> ( ord_less_eq_set_nat @ ( image_o_nat @ F @ A ) @ B4 ) ) ).
% image_subsetI
thf(fact_853_image__subsetI,axiom,
! [A: set_o,F: $o > $o,B4: set_o] :
( ! [X3: $o] :
( ( member_o @ X3 @ A )
=> ( member_o @ ( F @ X3 ) @ B4 ) )
=> ( ord_less_eq_set_o @ ( image_o_o @ F @ A ) @ B4 ) ) ).
% image_subsetI
thf(fact_854_image__subsetI,axiom,
! [A: set_real,F: real > real,B4: set_real] :
( ! [X3: real] :
( ( member_real @ X3 @ A )
=> ( member_real @ ( F @ X3 ) @ B4 ) )
=> ( ord_less_eq_set_real @ ( image_real_real @ F @ A ) @ B4 ) ) ).
% image_subsetI
thf(fact_855_image__subsetI,axiom,
! [A: set_nat,F: nat > real,B4: set_real] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( member_real @ ( F @ X3 ) @ B4 ) )
=> ( ord_less_eq_set_real @ ( image_nat_real @ F @ A ) @ B4 ) ) ).
% image_subsetI
thf(fact_856_image__subsetI,axiom,
! [A: set_o,F: $o > real,B4: set_real] :
( ! [X3: $o] :
( ( member_o @ X3 @ A )
=> ( member_real @ ( F @ X3 ) @ B4 ) )
=> ( ord_less_eq_set_real @ ( image_o_real @ F @ A ) @ B4 ) ) ).
% image_subsetI
thf(fact_857_image__mono,axiom,
! [A: set_nat,B4: set_nat,F: nat > int] :
( ( ord_less_eq_set_nat @ A @ B4 )
=> ( ord_less_eq_set_int @ ( image_nat_int @ F @ A ) @ ( image_nat_int @ F @ B4 ) ) ) ).
% image_mono
thf(fact_858_image__mono,axiom,
! [A: set_real,B4: set_real,F: real > real] :
( ( ord_less_eq_set_real @ A @ B4 )
=> ( ord_less_eq_set_real @ ( image_real_real @ F @ A ) @ ( image_real_real @ F @ B4 ) ) ) ).
% image_mono
thf(fact_859_less__eq__real__def,axiom,
( ord_less_eq_real
= ( ^ [X2: real,Y5: real] :
( ( ord_less_real @ X2 @ Y5 )
| ( X2 = Y5 ) ) ) ) ).
% less_eq_real_def
thf(fact_860_ex__gt__or__lt,axiom,
! [A2: real] :
? [B3: real] :
( ( ord_less_real @ A2 @ B3 )
| ( ord_less_real @ B3 @ A2 ) ) ).
% ex_gt_or_lt
thf(fact_861_field__lbound__gt__zero,axiom,
! [D1: real,D2: real] :
( ( ord_less_real @ zero_zero_real @ D1 )
=> ( ( ord_less_real @ zero_zero_real @ D2 )
=> ? [E3: real] :
( ( ord_less_real @ zero_zero_real @ E3 )
& ( ord_less_real @ E3 @ D1 )
& ( ord_less_real @ E3 @ D2 ) ) ) ) ).
% field_lbound_gt_zero
thf(fact_862_linorder__neqE__linordered__idom,axiom,
! [X: real,Y: real] :
( ( X != Y )
=> ( ~ ( ord_less_real @ X @ Y )
=> ( ord_less_real @ Y @ X ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_863_linorder__neqE__linordered__idom,axiom,
! [X: int,Y: int] :
( ( X != Y )
=> ( ~ ( ord_less_int @ X @ Y )
=> ( ord_less_int @ Y @ X ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_864_imageE,axiom,
! [B: int,F: nat > int,A: set_nat] :
( ( member_int @ B @ ( image_nat_int @ F @ A ) )
=> ~ ! [X3: nat] :
( ( B
= ( F @ X3 ) )
=> ~ ( member_nat @ X3 @ A ) ) ) ).
% imageE
thf(fact_865_imageE,axiom,
! [B: real,F: real > real,A: set_real] :
( ( member_real @ B @ ( image_real_real @ F @ A ) )
=> ~ ! [X3: real] :
( ( B
= ( F @ X3 ) )
=> ~ ( member_real @ X3 @ A ) ) ) ).
% imageE
thf(fact_866_imageE,axiom,
! [B: real,F: nat > real,A: set_nat] :
( ( member_real @ B @ ( image_nat_real @ F @ A ) )
=> ~ ! [X3: nat] :
( ( B
= ( F @ X3 ) )
=> ~ ( member_nat @ X3 @ A ) ) ) ).
% imageE
thf(fact_867_imageE,axiom,
! [B: real,F: $o > real,A: set_o] :
( ( member_real @ B @ ( image_o_real @ F @ A ) )
=> ~ ! [X3: $o] :
( ( B
= ( F @ X3 ) )
=> ~ ( member_o @ X3 @ A ) ) ) ).
% imageE
thf(fact_868_imageE,axiom,
! [B: nat,F: real > nat,A: set_real] :
( ( member_nat @ B @ ( image_real_nat @ F @ A ) )
=> ~ ! [X3: real] :
( ( B
= ( F @ X3 ) )
=> ~ ( member_real @ X3 @ A ) ) ) ).
% imageE
thf(fact_869_imageE,axiom,
! [B: nat,F: nat > nat,A: set_nat] :
( ( member_nat @ B @ ( image_nat_nat @ F @ A ) )
=> ~ ! [X3: nat] :
( ( B
= ( F @ X3 ) )
=> ~ ( member_nat @ X3 @ A ) ) ) ).
% imageE
thf(fact_870_imageE,axiom,
! [B: nat,F: $o > nat,A: set_o] :
( ( member_nat @ B @ ( image_o_nat @ F @ A ) )
=> ~ ! [X3: $o] :
( ( B
= ( F @ X3 ) )
=> ~ ( member_o @ X3 @ A ) ) ) ).
% imageE
thf(fact_871_imageE,axiom,
! [B: $o,F: real > $o,A: set_real] :
( ( member_o @ B @ ( image_real_o @ F @ A ) )
=> ~ ! [X3: real] :
( ( B
= ( F @ X3 ) )
=> ~ ( member_real @ X3 @ A ) ) ) ).
% imageE
thf(fact_872_imageE,axiom,
! [B: $o,F: nat > $o,A: set_nat] :
( ( member_o @ B @ ( image_nat_o @ F @ A ) )
=> ~ ! [X3: nat] :
( ( B
= ( F @ X3 ) )
=> ~ ( member_nat @ X3 @ A ) ) ) ).
% imageE
thf(fact_873_imageE,axiom,
! [B: $o,F: $o > $o,A: set_o] :
( ( member_o @ B @ ( image_o_o @ F @ A ) )
=> ~ ! [X3: $o] :
( ( B
= ( F @ X3 ) )
=> ~ ( member_o @ X3 @ A ) ) ) ).
% imageE
thf(fact_874_imageI,axiom,
! [X: real,A: set_real,F: real > real] :
( ( member_real @ X @ A )
=> ( member_real @ ( F @ X ) @ ( image_real_real @ F @ A ) ) ) ).
% imageI
thf(fact_875_imageI,axiom,
! [X: real,A: set_real,F: real > nat] :
( ( member_real @ X @ A )
=> ( member_nat @ ( F @ X ) @ ( image_real_nat @ F @ A ) ) ) ).
% imageI
thf(fact_876_imageI,axiom,
! [X: real,A: set_real,F: real > $o] :
( ( member_real @ X @ A )
=> ( member_o @ ( F @ X ) @ ( image_real_o @ F @ A ) ) ) ).
% imageI
thf(fact_877_imageI,axiom,
! [X: nat,A: set_nat,F: nat > int] :
( ( member_nat @ X @ A )
=> ( member_int @ ( F @ X ) @ ( image_nat_int @ F @ A ) ) ) ).
% imageI
thf(fact_878_imageI,axiom,
! [X: nat,A: set_nat,F: nat > real] :
( ( member_nat @ X @ A )
=> ( member_real @ ( F @ X ) @ ( image_nat_real @ F @ A ) ) ) ).
% imageI
thf(fact_879_imageI,axiom,
! [X: nat,A: set_nat,F: nat > nat] :
( ( member_nat @ X @ A )
=> ( member_nat @ ( F @ X ) @ ( image_nat_nat @ F @ A ) ) ) ).
% imageI
thf(fact_880_imageI,axiom,
! [X: nat,A: set_nat,F: nat > $o] :
( ( member_nat @ X @ A )
=> ( member_o @ ( F @ X ) @ ( image_nat_o @ F @ A ) ) ) ).
% imageI
thf(fact_881_imageI,axiom,
! [X: $o,A: set_o,F: $o > real] :
( ( member_o @ X @ A )
=> ( member_real @ ( F @ X ) @ ( image_o_real @ F @ A ) ) ) ).
% imageI
thf(fact_882_imageI,axiom,
! [X: $o,A: set_o,F: $o > nat] :
( ( member_o @ X @ A )
=> ( member_nat @ ( F @ X ) @ ( image_o_nat @ F @ A ) ) ) ).
% imageI
thf(fact_883_imageI,axiom,
! [X: $o,A: set_o,F: $o > $o] :
( ( member_o @ X @ A )
=> ( member_o @ ( F @ X ) @ ( image_o_o @ F @ A ) ) ) ).
% imageI
thf(fact_884_image__iff,axiom,
! [Z2: int,F: nat > int,A: set_nat] :
( ( member_int @ Z2 @ ( image_nat_int @ F @ A ) )
= ( ? [X2: nat] :
( ( member_nat @ X2 @ A )
& ( Z2
= ( F @ X2 ) ) ) ) ) ).
% image_iff
thf(fact_885_image__iff,axiom,
! [Z2: real,F: real > real,A: set_real] :
( ( member_real @ Z2 @ ( image_real_real @ F @ A ) )
= ( ? [X2: real] :
( ( member_real @ X2 @ A )
& ( Z2
= ( F @ X2 ) ) ) ) ) ).
% image_iff
thf(fact_886_bex__imageD,axiom,
! [F: real > real,A: set_real,P: real > $o] :
( ? [X4: real] :
( ( member_real @ X4 @ ( image_real_real @ F @ A ) )
& ( P @ X4 ) )
=> ? [X3: real] :
( ( member_real @ X3 @ A )
& ( P @ ( F @ X3 ) ) ) ) ).
% bex_imageD
thf(fact_887_bex__imageD,axiom,
! [F: nat > int,A: set_nat,P: int > $o] :
( ? [X4: int] :
( ( member_int @ X4 @ ( image_nat_int @ F @ A ) )
& ( P @ X4 ) )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A )
& ( P @ ( F @ X3 ) ) ) ) ).
% bex_imageD
thf(fact_888_image__cong,axiom,
! [M3: set_real,N4: set_real,F: real > real,G: real > real] :
( ( M3 = N4 )
=> ( ! [X3: real] :
( ( member_real @ X3 @ N4 )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( image_real_real @ F @ M3 )
= ( image_real_real @ G @ N4 ) ) ) ) ).
% image_cong
thf(fact_889_image__cong,axiom,
! [M3: set_nat,N4: set_nat,F: nat > int,G: nat > int] :
( ( M3 = N4 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ N4 )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( image_nat_int @ F @ M3 )
= ( image_nat_int @ G @ N4 ) ) ) ) ).
% image_cong
thf(fact_890_ball__imageD,axiom,
! [F: real > real,A: set_real,P: real > $o] :
( ! [X3: real] :
( ( member_real @ X3 @ ( image_real_real @ F @ A ) )
=> ( P @ X3 ) )
=> ! [X4: real] :
( ( member_real @ X4 @ A )
=> ( P @ ( F @ X4 ) ) ) ) ).
% ball_imageD
thf(fact_891_ball__imageD,axiom,
! [F: nat > int,A: set_nat,P: int > $o] :
( ! [X3: int] :
( ( member_int @ X3 @ ( image_nat_int @ F @ A ) )
=> ( P @ X3 ) )
=> ! [X4: nat] :
( ( member_nat @ X4 @ A )
=> ( P @ ( F @ X4 ) ) ) ) ).
% ball_imageD
thf(fact_892_image__image,axiom,
! [F: int > int,G: nat > int,A: set_nat] :
( ( image_int_int @ F @ ( image_nat_int @ G @ A ) )
= ( image_nat_int
@ ^ [X2: nat] : ( F @ ( G @ X2 ) )
@ A ) ) ).
% image_image
thf(fact_893_image__image,axiom,
! [F: real > real,G: real > real,A: set_real] :
( ( image_real_real @ F @ ( image_real_real @ G @ A ) )
= ( image_real_real
@ ^ [X2: real] : ( F @ ( G @ X2 ) )
@ A ) ) ).
% image_image
thf(fact_894_image__image,axiom,
! [F: nat > int,G: nat > nat,A: set_nat] :
( ( image_nat_int @ F @ ( image_nat_nat @ G @ A ) )
= ( image_nat_int
@ ^ [X2: nat] : ( F @ ( G @ X2 ) )
@ A ) ) ).
% image_image
thf(fact_895_rev__image__eqI,axiom,
! [X: real,A: set_real,B: real,F: real > real] :
( ( member_real @ X @ A )
=> ( ( B
= ( F @ X ) )
=> ( member_real @ B @ ( image_real_real @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_896_rev__image__eqI,axiom,
! [X: real,A: set_real,B: nat,F: real > nat] :
( ( member_real @ X @ A )
=> ( ( B
= ( F @ X ) )
=> ( member_nat @ B @ ( image_real_nat @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_897_rev__image__eqI,axiom,
! [X: real,A: set_real,B: $o,F: real > $o] :
( ( member_real @ X @ A )
=> ( ( B
= ( F @ X ) )
=> ( member_o @ B @ ( image_real_o @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_898_rev__image__eqI,axiom,
! [X: nat,A: set_nat,B: int,F: nat > int] :
( ( member_nat @ X @ A )
=> ( ( B
= ( F @ X ) )
=> ( member_int @ B @ ( image_nat_int @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_899_rev__image__eqI,axiom,
! [X: nat,A: set_nat,B: real,F: nat > real] :
( ( member_nat @ X @ A )
=> ( ( B
= ( F @ X ) )
=> ( member_real @ B @ ( image_nat_real @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_900_rev__image__eqI,axiom,
! [X: nat,A: set_nat,B: nat,F: nat > nat] :
( ( member_nat @ X @ A )
=> ( ( B
= ( F @ X ) )
=> ( member_nat @ B @ ( image_nat_nat @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_901_rev__image__eqI,axiom,
! [X: nat,A: set_nat,B: $o,F: nat > $o] :
( ( member_nat @ X @ A )
=> ( ( B
= ( F @ X ) )
=> ( member_o @ B @ ( image_nat_o @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_902_rev__image__eqI,axiom,
! [X: $o,A: set_o,B: real,F: $o > real] :
( ( member_o @ X @ A )
=> ( ( B
= ( F @ X ) )
=> ( member_real @ B @ ( image_o_real @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_903_rev__image__eqI,axiom,
! [X: $o,A: set_o,B: nat,F: $o > nat] :
( ( member_o @ X @ A )
=> ( ( B
= ( F @ X ) )
=> ( member_nat @ B @ ( image_o_nat @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_904_rev__image__eqI,axiom,
! [X: $o,A: set_o,B: $o,F: $o > $o] :
( ( member_o @ X @ A )
=> ( ( B
= ( F @ X ) )
=> ( member_o @ B @ ( image_o_o @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_905_Compr__image__eq,axiom,
! [F: nat > int,A: set_nat,P: int > $o] :
( ( collect_int
@ ^ [X2: int] :
( ( member_int @ X2 @ ( image_nat_int @ F @ A ) )
& ( P @ X2 ) ) )
= ( image_nat_int @ F
@ ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ A )
& ( P @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_906_Compr__image__eq,axiom,
! [F: real > real,A: set_real,P: real > $o] :
( ( collect_real
@ ^ [X2: real] :
( ( member_real @ X2 @ ( image_real_real @ F @ A ) )
& ( P @ X2 ) ) )
= ( image_real_real @ F
@ ( collect_real
@ ^ [X2: real] :
( ( member_real @ X2 @ A )
& ( P @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_907_Compr__image__eq,axiom,
! [F: nat > real,A: set_nat,P: real > $o] :
( ( collect_real
@ ^ [X2: real] :
( ( member_real @ X2 @ ( image_nat_real @ F @ A ) )
& ( P @ X2 ) ) )
= ( image_nat_real @ F
@ ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ A )
& ( P @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_908_Compr__image__eq,axiom,
! [F: $o > real,A: set_o,P: real > $o] :
( ( collect_real
@ ^ [X2: real] :
( ( member_real @ X2 @ ( image_o_real @ F @ A ) )
& ( P @ X2 ) ) )
= ( image_o_real @ F
@ ( collect_o
@ ^ [X2: $o] :
( ( member_o @ X2 @ A )
& ( P @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_909_Compr__image__eq,axiom,
! [F: real > nat,A: set_real,P: nat > $o] :
( ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ ( image_real_nat @ F @ A ) )
& ( P @ X2 ) ) )
= ( image_real_nat @ F
@ ( collect_real
@ ^ [X2: real] :
( ( member_real @ X2 @ A )
& ( P @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_910_Compr__image__eq,axiom,
! [F: nat > nat,A: set_nat,P: nat > $o] :
( ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ ( image_nat_nat @ F @ A ) )
& ( P @ X2 ) ) )
= ( image_nat_nat @ F
@ ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ A )
& ( P @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_911_Compr__image__eq,axiom,
! [F: $o > nat,A: set_o,P: nat > $o] :
( ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ ( image_o_nat @ F @ A ) )
& ( P @ X2 ) ) )
= ( image_o_nat @ F
@ ( collect_o
@ ^ [X2: $o] :
( ( member_o @ X2 @ A )
& ( P @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_912_Compr__image__eq,axiom,
! [F: real > $o,A: set_real,P: $o > $o] :
( ( collect_o
@ ^ [X2: $o] :
( ( member_o @ X2 @ ( image_real_o @ F @ A ) )
& ( P @ X2 ) ) )
= ( image_real_o @ F
@ ( collect_real
@ ^ [X2: real] :
( ( member_real @ X2 @ A )
& ( P @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_913_Compr__image__eq,axiom,
! [F: nat > $o,A: set_nat,P: $o > $o] :
( ( collect_o
@ ^ [X2: $o] :
( ( member_o @ X2 @ ( image_nat_o @ F @ A ) )
& ( P @ X2 ) ) )
= ( image_nat_o @ F
@ ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ A )
& ( P @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_914_Compr__image__eq,axiom,
! [F: $o > $o,A: set_o,P: $o > $o] :
( ( collect_o
@ ^ [X2: $o] :
( ( member_o @ X2 @ ( image_o_o @ F @ A ) )
& ( P @ X2 ) ) )
= ( image_o_o @ F
@ ( collect_o
@ ^ [X2: $o] :
( ( member_o @ X2 @ A )
& ( P @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_915_linordered__field__no__ub,axiom,
! [X4: real] :
? [X_1: real] : ( ord_less_real @ X4 @ X_1 ) ).
% linordered_field_no_ub
thf(fact_916_linordered__field__no__lb,axiom,
! [X4: real] :
? [Y2: real] : ( ord_less_real @ Y2 @ X4 ) ).
% linordered_field_no_lb
thf(fact_917_reals__Archimedean2,axiom,
! [X: real] :
? [N3: nat] : ( ord_less_real @ X @ ( semiri5074537144036343181t_real @ N3 ) ) ).
% reals_Archimedean2
thf(fact_918_Sup_OSUP__identity__eq,axiom,
! [Sup: set_real > real,A: set_real] :
( ( Sup
@ ( image_real_real
@ ^ [X2: real] : X2
@ A ) )
= ( Sup @ A ) ) ).
% Sup.SUP_identity_eq
thf(fact_919_Inf_OINF__identity__eq,axiom,
! [Inf: set_real > real,A: set_real] :
( ( Inf
@ ( image_real_real
@ ^ [X2: real] : X2
@ A ) )
= ( Inf @ A ) ) ).
% Inf.INF_identity_eq
thf(fact_920_Sup_OSUP__cong,axiom,
! [A: set_real,B4: set_real,C2: real > real,D3: real > real,Sup: set_real > real] :
( ( A = B4 )
=> ( ! [X3: real] :
( ( member_real @ X3 @ B4 )
=> ( ( C2 @ X3 )
= ( D3 @ X3 ) ) )
=> ( ( Sup @ ( image_real_real @ C2 @ A ) )
= ( Sup @ ( image_real_real @ D3 @ B4 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_921_Sup_OSUP__cong,axiom,
! [A: set_nat,B4: set_nat,C2: nat > int,D3: nat > int,Sup: set_int > int] :
( ( A = B4 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ B4 )
=> ( ( C2 @ X3 )
= ( D3 @ X3 ) ) )
=> ( ( Sup @ ( image_nat_int @ C2 @ A ) )
= ( Sup @ ( image_nat_int @ D3 @ B4 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_922_Inf_OINF__cong,axiom,
! [A: set_real,B4: set_real,C2: real > real,D3: real > real,Inf: set_real > real] :
( ( A = B4 )
=> ( ! [X3: real] :
( ( member_real @ X3 @ B4 )
=> ( ( C2 @ X3 )
= ( D3 @ X3 ) ) )
=> ( ( Inf @ ( image_real_real @ C2 @ A ) )
= ( Inf @ ( image_real_real @ D3 @ B4 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_923_Inf_OINF__cong,axiom,
! [A: set_nat,B4: set_nat,C2: nat > int,D3: nat > int,Inf: set_int > int] :
( ( A = B4 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ B4 )
=> ( ( C2 @ X3 )
= ( D3 @ X3 ) ) )
=> ( ( Inf @ ( image_nat_int @ C2 @ A ) )
= ( Inf @ ( image_nat_int @ D3 @ B4 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_924_less__imp__of__nat__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% less_imp_of_nat_less
thf(fact_925_less__imp__of__nat__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).
% less_imp_of_nat_less
thf(fact_926_less__imp__of__nat__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% less_imp_of_nat_less
thf(fact_927_of__nat__less__imp__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
=> ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_imp_less
thf(fact_928_of__nat__less__imp__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) )
=> ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_imp_less
thf(fact_929_of__nat__less__imp__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
=> ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_imp_less
thf(fact_930_atLeastatMost__psubset__iff,axiom,
! [A2: set_real,B: set_real,C: set_real,D: set_real] :
( ( ord_le7926960851185191020t_real @ ( set_or7743017856606604397t_real @ A2 @ B ) @ ( set_or7743017856606604397t_real @ C @ D ) )
= ( ( ~ ( ord_less_eq_set_real @ A2 @ B )
| ( ( ord_less_eq_set_real @ C @ A2 )
& ( ord_less_eq_set_real @ B @ D )
& ( ( ord_less_set_real @ C @ A2 )
| ( ord_less_set_real @ B @ D ) ) ) )
& ( ord_less_eq_set_real @ C @ D ) ) ) ).
% atLeastatMost_psubset_iff
thf(fact_931_atLeastatMost__psubset__iff,axiom,
! [A2: real,B: real,C: real,D: real] :
( ( ord_less_set_real @ ( set_or1222579329274155063t_real @ A2 @ B ) @ ( set_or1222579329274155063t_real @ C @ D ) )
= ( ( ~ ( ord_less_eq_real @ A2 @ B )
| ( ( ord_less_eq_real @ C @ A2 )
& ( ord_less_eq_real @ B @ D )
& ( ( ord_less_real @ C @ A2 )
| ( ord_less_real @ B @ D ) ) ) )
& ( ord_less_eq_real @ C @ D ) ) ) ).
% atLeastatMost_psubset_iff
thf(fact_932_atLeastatMost__psubset__iff,axiom,
! [A2: nat,B: nat,C: nat,D: nat] :
( ( ord_less_set_nat @ ( set_or1269000886237332187st_nat @ A2 @ B ) @ ( set_or1269000886237332187st_nat @ C @ D ) )
= ( ( ~ ( ord_less_eq_nat @ A2 @ B )
| ( ( ord_less_eq_nat @ C @ A2 )
& ( ord_less_eq_nat @ B @ D )
& ( ( ord_less_nat @ C @ A2 )
| ( ord_less_nat @ B @ D ) ) ) )
& ( ord_less_eq_nat @ C @ D ) ) ) ).
% atLeastatMost_psubset_iff
thf(fact_933_atLeastatMost__psubset__iff,axiom,
! [A2: int,B: int,C: int,D: int] :
( ( ord_less_set_int @ ( set_or1266510415728281911st_int @ A2 @ B ) @ ( set_or1266510415728281911st_int @ C @ D ) )
= ( ( ~ ( ord_less_eq_int @ A2 @ B )
| ( ( ord_less_eq_int @ C @ A2 )
& ( ord_less_eq_int @ B @ D )
& ( ( ord_less_int @ C @ A2 )
| ( ord_less_int @ B @ D ) ) ) )
& ( ord_less_eq_int @ C @ D ) ) ) ).
% atLeastatMost_psubset_iff
thf(fact_934_SUP__cong,axiom,
! [A: set_nat,B4: set_nat,C2: nat > int,D3: nat > int] :
( ( A = B4 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ B4 )
=> ( ( C2 @ X3 )
= ( D3 @ X3 ) ) )
=> ( ( complete_Sup_Sup_int @ ( image_nat_int @ C2 @ A ) )
= ( complete_Sup_Sup_int @ ( image_nat_int @ D3 @ B4 ) ) ) ) ) ).
% SUP_cong
thf(fact_935_SUP__cong,axiom,
! [A: set_real,B4: set_real,C2: real > real,D3: real > real] :
( ( A = B4 )
=> ( ! [X3: real] :
( ( member_real @ X3 @ B4 )
=> ( ( C2 @ X3 )
= ( D3 @ X3 ) ) )
=> ( ( comple1385675409528146559p_real @ ( image_real_real @ C2 @ A ) )
= ( comple1385675409528146559p_real @ ( image_real_real @ D3 @ B4 ) ) ) ) ) ).
% SUP_cong
thf(fact_936_SUP__cong,axiom,
! [A: set_nat,B4: set_nat,C2: nat > real,D3: nat > real] :
( ( A = B4 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ B4 )
=> ( ( C2 @ X3 )
= ( D3 @ X3 ) ) )
=> ( ( comple1385675409528146559p_real @ ( image_nat_real @ C2 @ A ) )
= ( comple1385675409528146559p_real @ ( image_nat_real @ D3 @ B4 ) ) ) ) ) ).
% SUP_cong
thf(fact_937_SUP__cong,axiom,
! [A: set_o,B4: set_o,C2: $o > real,D3: $o > real] :
( ( A = B4 )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ B4 )
=> ( ( C2 @ X3 )
= ( D3 @ X3 ) ) )
=> ( ( comple1385675409528146559p_real @ ( image_o_real @ C2 @ A ) )
= ( comple1385675409528146559p_real @ ( image_o_real @ D3 @ B4 ) ) ) ) ) ).
% SUP_cong
thf(fact_938_SUP__cong,axiom,
! [A: set_real,B4: set_real,C2: real > nat,D3: real > nat] :
( ( A = B4 )
=> ( ! [X3: real] :
( ( member_real @ X3 @ B4 )
=> ( ( C2 @ X3 )
= ( D3 @ X3 ) ) )
=> ( ( complete_Sup_Sup_nat @ ( image_real_nat @ C2 @ A ) )
= ( complete_Sup_Sup_nat @ ( image_real_nat @ D3 @ B4 ) ) ) ) ) ).
% SUP_cong
thf(fact_939_SUP__cong,axiom,
! [A: set_nat,B4: set_nat,C2: nat > nat,D3: nat > nat] :
( ( A = B4 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ B4 )
=> ( ( C2 @ X3 )
= ( D3 @ X3 ) ) )
=> ( ( complete_Sup_Sup_nat @ ( image_nat_nat @ C2 @ A ) )
= ( complete_Sup_Sup_nat @ ( image_nat_nat @ D3 @ B4 ) ) ) ) ) ).
% SUP_cong
thf(fact_940_SUP__cong,axiom,
! [A: set_o,B4: set_o,C2: $o > nat,D3: $o > nat] :
( ( A = B4 )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ B4 )
=> ( ( C2 @ X3 )
= ( D3 @ X3 ) ) )
=> ( ( complete_Sup_Sup_nat @ ( image_o_nat @ C2 @ A ) )
= ( complete_Sup_Sup_nat @ ( image_o_nat @ D3 @ B4 ) ) ) ) ) ).
% SUP_cong
thf(fact_941_SUP__cong,axiom,
! [A: set_real,B4: set_real,C2: real > $o,D3: real > $o] :
( ( A = B4 )
=> ( ! [X3: real] :
( ( member_real @ X3 @ B4 )
=> ( ( C2 @ X3 )
= ( D3 @ X3 ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_real_o @ C2 @ A ) )
= ( complete_Sup_Sup_o @ ( image_real_o @ D3 @ B4 ) ) ) ) ) ).
% SUP_cong
thf(fact_942_SUP__cong,axiom,
! [A: set_nat,B4: set_nat,C2: nat > $o,D3: nat > $o] :
( ( A = B4 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ B4 )
=> ( ( C2 @ X3 )
= ( D3 @ X3 ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_nat_o @ C2 @ A ) )
= ( complete_Sup_Sup_o @ ( image_nat_o @ D3 @ B4 ) ) ) ) ) ).
% SUP_cong
thf(fact_943_SUP__cong,axiom,
! [A: set_o,B4: set_o,C2: $o > $o,D3: $o > $o] :
( ( A = B4 )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ B4 )
=> ( ( C2 @ X3 )
= ( D3 @ X3 ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_o_o @ C2 @ A ) )
= ( complete_Sup_Sup_o @ ( image_o_o @ D3 @ B4 ) ) ) ) ) ).
% SUP_cong
thf(fact_944_complete__interval,axiom,
! [A2: real,B: real,P: real > $o] :
( ( ord_less_real @ A2 @ B )
=> ( ( P @ A2 )
=> ( ~ ( P @ B )
=> ? [C5: real] :
( ( ord_less_eq_real @ A2 @ C5 )
& ( ord_less_eq_real @ C5 @ B )
& ! [X4: real] :
( ( ( ord_less_eq_real @ A2 @ X4 )
& ( ord_less_real @ X4 @ C5 ) )
=> ( P @ X4 ) )
& ! [D4: real] :
( ! [X3: real] :
( ( ( ord_less_eq_real @ A2 @ X3 )
& ( ord_less_real @ X3 @ D4 ) )
=> ( P @ X3 ) )
=> ( ord_less_eq_real @ D4 @ C5 ) ) ) ) ) ) ).
% complete_interval
thf(fact_945_complete__interval,axiom,
! [A2: nat,B: nat,P: nat > $o] :
( ( ord_less_nat @ A2 @ B )
=> ( ( P @ A2 )
=> ( ~ ( P @ B )
=> ? [C5: nat] :
( ( ord_less_eq_nat @ A2 @ C5 )
& ( ord_less_eq_nat @ C5 @ B )
& ! [X4: nat] :
( ( ( ord_less_eq_nat @ A2 @ X4 )
& ( ord_less_nat @ X4 @ C5 ) )
=> ( P @ X4 ) )
& ! [D4: nat] :
( ! [X3: nat] :
( ( ( ord_less_eq_nat @ A2 @ X3 )
& ( ord_less_nat @ X3 @ D4 ) )
=> ( P @ X3 ) )
=> ( ord_less_eq_nat @ D4 @ C5 ) ) ) ) ) ) ).
% complete_interval
thf(fact_946_complete__interval,axiom,
! [A2: int,B: int,P: int > $o] :
( ( ord_less_int @ A2 @ B )
=> ( ( P @ A2 )
=> ( ~ ( P @ B )
=> ? [C5: int] :
( ( ord_less_eq_int @ A2 @ C5 )
& ( ord_less_eq_int @ C5 @ B )
& ! [X4: int] :
( ( ( ord_less_eq_int @ A2 @ X4 )
& ( ord_less_int @ X4 @ C5 ) )
=> ( P @ X4 ) )
& ! [D4: int] :
( ! [X3: int] :
( ( ( ord_less_eq_int @ A2 @ X3 )
& ( ord_less_int @ X3 @ D4 ) )
=> ( P @ X3 ) )
=> ( ord_less_eq_int @ D4 @ C5 ) ) ) ) ) ) ).
% complete_interval
thf(fact_947_eucl__less__le__not__le,axiom,
( ord_less_real
= ( ^ [X2: real,Y5: real] :
( ( ord_less_eq_real @ X2 @ Y5 )
& ~ ( ord_less_eq_real @ Y5 @ X2 ) ) ) ) ).
% eucl_less_le_not_le
thf(fact_948_INF__cong,axiom,
! [A: set_nat,B4: set_nat,C2: nat > int,D3: nat > int] :
( ( A = B4 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ B4 )
=> ( ( C2 @ X3 )
= ( D3 @ X3 ) ) )
=> ( ( complete_Inf_Inf_int @ ( image_nat_int @ C2 @ A ) )
= ( complete_Inf_Inf_int @ ( image_nat_int @ D3 @ B4 ) ) ) ) ) ).
% INF_cong
thf(fact_949_INF__cong,axiom,
! [A: set_real,B4: set_real,C2: real > real,D3: real > real] :
( ( A = B4 )
=> ( ! [X3: real] :
( ( member_real @ X3 @ B4 )
=> ( ( C2 @ X3 )
= ( D3 @ X3 ) ) )
=> ( ( comple4887499456419720421f_real @ ( image_real_real @ C2 @ A ) )
= ( comple4887499456419720421f_real @ ( image_real_real @ D3 @ B4 ) ) ) ) ) ).
% INF_cong
thf(fact_950_INF__cong,axiom,
! [A: set_nat,B4: set_nat,C2: nat > real,D3: nat > real] :
( ( A = B4 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ B4 )
=> ( ( C2 @ X3 )
= ( D3 @ X3 ) ) )
=> ( ( comple4887499456419720421f_real @ ( image_nat_real @ C2 @ A ) )
= ( comple4887499456419720421f_real @ ( image_nat_real @ D3 @ B4 ) ) ) ) ) ).
% INF_cong
thf(fact_951_INF__cong,axiom,
! [A: set_o,B4: set_o,C2: $o > real,D3: $o > real] :
( ( A = B4 )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ B4 )
=> ( ( C2 @ X3 )
= ( D3 @ X3 ) ) )
=> ( ( comple4887499456419720421f_real @ ( image_o_real @ C2 @ A ) )
= ( comple4887499456419720421f_real @ ( image_o_real @ D3 @ B4 ) ) ) ) ) ).
% INF_cong
thf(fact_952_INF__cong,axiom,
! [A: set_real,B4: set_real,C2: real > nat,D3: real > nat] :
( ( A = B4 )
=> ( ! [X3: real] :
( ( member_real @ X3 @ B4 )
=> ( ( C2 @ X3 )
= ( D3 @ X3 ) ) )
=> ( ( complete_Inf_Inf_nat @ ( image_real_nat @ C2 @ A ) )
= ( complete_Inf_Inf_nat @ ( image_real_nat @ D3 @ B4 ) ) ) ) ) ).
% INF_cong
thf(fact_953_INF__cong,axiom,
! [A: set_nat,B4: set_nat,C2: nat > nat,D3: nat > nat] :
( ( A = B4 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ B4 )
=> ( ( C2 @ X3 )
= ( D3 @ X3 ) ) )
=> ( ( complete_Inf_Inf_nat @ ( image_nat_nat @ C2 @ A ) )
= ( complete_Inf_Inf_nat @ ( image_nat_nat @ D3 @ B4 ) ) ) ) ) ).
% INF_cong
thf(fact_954_INF__cong,axiom,
! [A: set_o,B4: set_o,C2: $o > nat,D3: $o > nat] :
( ( A = B4 )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ B4 )
=> ( ( C2 @ X3 )
= ( D3 @ X3 ) ) )
=> ( ( complete_Inf_Inf_nat @ ( image_o_nat @ C2 @ A ) )
= ( complete_Inf_Inf_nat @ ( image_o_nat @ D3 @ B4 ) ) ) ) ) ).
% INF_cong
thf(fact_955_INF__cong,axiom,
! [A: set_real,B4: set_real,C2: real > $o,D3: real > $o] :
( ( A = B4 )
=> ( ! [X3: real] :
( ( member_real @ X3 @ B4 )
=> ( ( C2 @ X3 )
= ( D3 @ X3 ) ) )
=> ( ( complete_Inf_Inf_o @ ( image_real_o @ C2 @ A ) )
= ( complete_Inf_Inf_o @ ( image_real_o @ D3 @ B4 ) ) ) ) ) ).
% INF_cong
thf(fact_956_INF__cong,axiom,
! [A: set_nat,B4: set_nat,C2: nat > $o,D3: nat > $o] :
( ( A = B4 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ B4 )
=> ( ( C2 @ X3 )
= ( D3 @ X3 ) ) )
=> ( ( complete_Inf_Inf_o @ ( image_nat_o @ C2 @ A ) )
= ( complete_Inf_Inf_o @ ( image_nat_o @ D3 @ B4 ) ) ) ) ) ).
% INF_cong
thf(fact_957_INF__cong,axiom,
! [A: set_o,B4: set_o,C2: $o > $o,D3: $o > $o] :
( ( A = B4 )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ B4 )
=> ( ( C2 @ X3 )
= ( D3 @ X3 ) ) )
=> ( ( complete_Inf_Inf_o @ ( image_o_o @ C2 @ A ) )
= ( complete_Inf_Inf_o @ ( image_o_o @ D3 @ B4 ) ) ) ) ) ).
% INF_cong
thf(fact_958_zero__less__iff__neq__zero,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( N != zero_zero_nat ) ) ).
% zero_less_iff_neq_zero
thf(fact_959_gr__implies__not__zero,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_960_not__less__zero,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less_zero
thf(fact_961_gr__zeroI,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr_zeroI
thf(fact_962_diff__strict__right__mono,axiom,
! [A2: real,B: real,C: real] :
( ( ord_less_real @ A2 @ B )
=> ( ord_less_real @ ( minus_minus_real @ A2 @ C ) @ ( minus_minus_real @ B @ C ) ) ) ).
% diff_strict_right_mono
thf(fact_963_diff__strict__right__mono,axiom,
! [A2: int,B: int,C: int] :
( ( ord_less_int @ A2 @ B )
=> ( ord_less_int @ ( minus_minus_int @ A2 @ C ) @ ( minus_minus_int @ B @ C ) ) ) ).
% diff_strict_right_mono
thf(fact_964_diff__strict__left__mono,axiom,
! [B: real,A2: real,C: real] :
( ( ord_less_real @ B @ A2 )
=> ( ord_less_real @ ( minus_minus_real @ C @ A2 ) @ ( minus_minus_real @ C @ B ) ) ) ).
% diff_strict_left_mono
thf(fact_965_diff__strict__left__mono,axiom,
! [B: int,A2: int,C: int] :
( ( ord_less_int @ B @ A2 )
=> ( ord_less_int @ ( minus_minus_int @ C @ A2 ) @ ( minus_minus_int @ C @ B ) ) ) ).
% diff_strict_left_mono
thf(fact_966_diff__eq__diff__less,axiom,
! [A2: real,B: real,C: real,D: real] :
( ( ( minus_minus_real @ A2 @ B )
= ( minus_minus_real @ C @ D ) )
=> ( ( ord_less_real @ A2 @ B )
= ( ord_less_real @ C @ D ) ) ) ).
% diff_eq_diff_less
thf(fact_967_diff__eq__diff__less,axiom,
! [A2: int,B: int,C: int,D: int] :
( ( ( minus_minus_int @ A2 @ B )
= ( minus_minus_int @ C @ D ) )
=> ( ( ord_less_int @ A2 @ B )
= ( ord_less_int @ C @ D ) ) ) ).
% diff_eq_diff_less
thf(fact_968_diff__strict__mono,axiom,
! [A2: real,B: real,D: real,C: real] :
( ( ord_less_real @ A2 @ B )
=> ( ( ord_less_real @ D @ C )
=> ( ord_less_real @ ( minus_minus_real @ A2 @ C ) @ ( minus_minus_real @ B @ D ) ) ) ) ).
% diff_strict_mono
thf(fact_969_diff__strict__mono,axiom,
! [A2: int,B: int,D: int,C: int] :
( ( ord_less_int @ A2 @ B )
=> ( ( ord_less_int @ D @ C )
=> ( ord_less_int @ ( minus_minus_int @ A2 @ C ) @ ( minus_minus_int @ B @ D ) ) ) ) ).
% diff_strict_mono
thf(fact_970_SUP__eq,axiom,
! [A: set_real,B4: set_real,F: real > $o,G: real > $o] :
( ! [I3: real] :
( ( member_real @ I3 @ A )
=> ? [X4: real] :
( ( member_real @ X4 @ B4 )
& ( ord_less_eq_o @ ( F @ I3 ) @ ( G @ X4 ) ) ) )
=> ( ! [J3: real] :
( ( member_real @ J3 @ B4 )
=> ? [X4: real] :
( ( member_real @ X4 @ A )
& ( ord_less_eq_o @ ( G @ J3 ) @ ( F @ X4 ) ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_real_o @ F @ A ) )
= ( complete_Sup_Sup_o @ ( image_real_o @ G @ B4 ) ) ) ) ) ).
% SUP_eq
thf(fact_971_SUP__eq,axiom,
! [A: set_real,B4: set_nat,F: real > $o,G: nat > $o] :
( ! [I3: real] :
( ( member_real @ I3 @ A )
=> ? [X4: nat] :
( ( member_nat @ X4 @ B4 )
& ( ord_less_eq_o @ ( F @ I3 ) @ ( G @ X4 ) ) ) )
=> ( ! [J3: nat] :
( ( member_nat @ J3 @ B4 )
=> ? [X4: real] :
( ( member_real @ X4 @ A )
& ( ord_less_eq_o @ ( G @ J3 ) @ ( F @ X4 ) ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_real_o @ F @ A ) )
= ( complete_Sup_Sup_o @ ( image_nat_o @ G @ B4 ) ) ) ) ) ).
% SUP_eq
thf(fact_972_SUP__eq,axiom,
! [A: set_real,B4: set_o,F: real > $o,G: $o > $o] :
( ! [I3: real] :
( ( member_real @ I3 @ A )
=> ? [X4: $o] :
( ( member_o @ X4 @ B4 )
& ( ord_less_eq_o @ ( F @ I3 ) @ ( G @ X4 ) ) ) )
=> ( ! [J3: $o] :
( ( member_o @ J3 @ B4 )
=> ? [X4: real] :
( ( member_real @ X4 @ A )
& ( ord_less_eq_o @ ( G @ J3 ) @ ( F @ X4 ) ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_real_o @ F @ A ) )
= ( complete_Sup_Sup_o @ ( image_o_o @ G @ B4 ) ) ) ) ) ).
% SUP_eq
thf(fact_973_SUP__eq,axiom,
! [A: set_nat,B4: set_real,F: nat > $o,G: real > $o] :
( ! [I3: nat] :
( ( member_nat @ I3 @ A )
=> ? [X4: real] :
( ( member_real @ X4 @ B4 )
& ( ord_less_eq_o @ ( F @ I3 ) @ ( G @ X4 ) ) ) )
=> ( ! [J3: real] :
( ( member_real @ J3 @ B4 )
=> ? [X4: nat] :
( ( member_nat @ X4 @ A )
& ( ord_less_eq_o @ ( G @ J3 ) @ ( F @ X4 ) ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_nat_o @ F @ A ) )
= ( complete_Sup_Sup_o @ ( image_real_o @ G @ B4 ) ) ) ) ) ).
% SUP_eq
thf(fact_974_SUP__eq,axiom,
! [A: set_nat,B4: set_nat,F: nat > $o,G: nat > $o] :
( ! [I3: nat] :
( ( member_nat @ I3 @ A )
=> ? [X4: nat] :
( ( member_nat @ X4 @ B4 )
& ( ord_less_eq_o @ ( F @ I3 ) @ ( G @ X4 ) ) ) )
=> ( ! [J3: nat] :
( ( member_nat @ J3 @ B4 )
=> ? [X4: nat] :
( ( member_nat @ X4 @ A )
& ( ord_less_eq_o @ ( G @ J3 ) @ ( F @ X4 ) ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_nat_o @ F @ A ) )
= ( complete_Sup_Sup_o @ ( image_nat_o @ G @ B4 ) ) ) ) ) ).
% SUP_eq
thf(fact_975_SUP__eq,axiom,
! [A: set_nat,B4: set_o,F: nat > $o,G: $o > $o] :
( ! [I3: nat] :
( ( member_nat @ I3 @ A )
=> ? [X4: $o] :
( ( member_o @ X4 @ B4 )
& ( ord_less_eq_o @ ( F @ I3 ) @ ( G @ X4 ) ) ) )
=> ( ! [J3: $o] :
( ( member_o @ J3 @ B4 )
=> ? [X4: nat] :
( ( member_nat @ X4 @ A )
& ( ord_less_eq_o @ ( G @ J3 ) @ ( F @ X4 ) ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_nat_o @ F @ A ) )
= ( complete_Sup_Sup_o @ ( image_o_o @ G @ B4 ) ) ) ) ) ).
% SUP_eq
thf(fact_976_SUP__eq,axiom,
! [A: set_o,B4: set_real,F: $o > $o,G: real > $o] :
( ! [I3: $o] :
( ( member_o @ I3 @ A )
=> ? [X4: real] :
( ( member_real @ X4 @ B4 )
& ( ord_less_eq_o @ ( F @ I3 ) @ ( G @ X4 ) ) ) )
=> ( ! [J3: real] :
( ( member_real @ J3 @ B4 )
=> ? [X4: $o] :
( ( member_o @ X4 @ A )
& ( ord_less_eq_o @ ( G @ J3 ) @ ( F @ X4 ) ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_o_o @ F @ A ) )
= ( complete_Sup_Sup_o @ ( image_real_o @ G @ B4 ) ) ) ) ) ).
% SUP_eq
thf(fact_977_SUP__eq,axiom,
! [A: set_o,B4: set_nat,F: $o > $o,G: nat > $o] :
( ! [I3: $o] :
( ( member_o @ I3 @ A )
=> ? [X4: nat] :
( ( member_nat @ X4 @ B4 )
& ( ord_less_eq_o @ ( F @ I3 ) @ ( G @ X4 ) ) ) )
=> ( ! [J3: nat] :
( ( member_nat @ J3 @ B4 )
=> ? [X4: $o] :
( ( member_o @ X4 @ A )
& ( ord_less_eq_o @ ( G @ J3 ) @ ( F @ X4 ) ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_o_o @ F @ A ) )
= ( complete_Sup_Sup_o @ ( image_nat_o @ G @ B4 ) ) ) ) ) ).
% SUP_eq
thf(fact_978_SUP__eq,axiom,
! [A: set_o,B4: set_o,F: $o > $o,G: $o > $o] :
( ! [I3: $o] :
( ( member_o @ I3 @ A )
=> ? [X4: $o] :
( ( member_o @ X4 @ B4 )
& ( ord_less_eq_o @ ( F @ I3 ) @ ( G @ X4 ) ) ) )
=> ( ! [J3: $o] :
( ( member_o @ J3 @ B4 )
=> ? [X4: $o] :
( ( member_o @ X4 @ A )
& ( ord_less_eq_o @ ( G @ J3 ) @ ( F @ X4 ) ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_o_o @ F @ A ) )
= ( complete_Sup_Sup_o @ ( image_o_o @ G @ B4 ) ) ) ) ) ).
% SUP_eq
thf(fact_979_SUP__eq,axiom,
! [A: set_real,B4: set_real,F: real > set_real,G: real > set_real] :
( ! [I3: real] :
( ( member_real @ I3 @ A )
=> ? [X4: real] :
( ( member_real @ X4 @ B4 )
& ( ord_less_eq_set_real @ ( F @ I3 ) @ ( G @ X4 ) ) ) )
=> ( ! [J3: real] :
( ( member_real @ J3 @ B4 )
=> ? [X4: real] :
( ( member_real @ X4 @ A )
& ( ord_less_eq_set_real @ ( G @ J3 ) @ ( F @ X4 ) ) ) )
=> ( ( comple3096694443085538997t_real @ ( image_real_set_real @ F @ A ) )
= ( comple3096694443085538997t_real @ ( image_real_set_real @ G @ B4 ) ) ) ) ) ).
% SUP_eq
thf(fact_980_INF__eq,axiom,
! [A: set_real,B4: set_real,G: real > $o,F: real > $o] :
( ! [I3: real] :
( ( member_real @ I3 @ A )
=> ? [X4: real] :
( ( member_real @ X4 @ B4 )
& ( ord_less_eq_o @ ( G @ X4 ) @ ( F @ I3 ) ) ) )
=> ( ! [J3: real] :
( ( member_real @ J3 @ B4 )
=> ? [X4: real] :
( ( member_real @ X4 @ A )
& ( ord_less_eq_o @ ( F @ X4 ) @ ( G @ J3 ) ) ) )
=> ( ( complete_Inf_Inf_o @ ( image_real_o @ F @ A ) )
= ( complete_Inf_Inf_o @ ( image_real_o @ G @ B4 ) ) ) ) ) ).
% INF_eq
thf(fact_981_INF__eq,axiom,
! [A: set_real,B4: set_nat,G: nat > $o,F: real > $o] :
( ! [I3: real] :
( ( member_real @ I3 @ A )
=> ? [X4: nat] :
( ( member_nat @ X4 @ B4 )
& ( ord_less_eq_o @ ( G @ X4 ) @ ( F @ I3 ) ) ) )
=> ( ! [J3: nat] :
( ( member_nat @ J3 @ B4 )
=> ? [X4: real] :
( ( member_real @ X4 @ A )
& ( ord_less_eq_o @ ( F @ X4 ) @ ( G @ J3 ) ) ) )
=> ( ( complete_Inf_Inf_o @ ( image_real_o @ F @ A ) )
= ( complete_Inf_Inf_o @ ( image_nat_o @ G @ B4 ) ) ) ) ) ).
% INF_eq
thf(fact_982_INF__eq,axiom,
! [A: set_real,B4: set_o,G: $o > $o,F: real > $o] :
( ! [I3: real] :
( ( member_real @ I3 @ A )
=> ? [X4: $o] :
( ( member_o @ X4 @ B4 )
& ( ord_less_eq_o @ ( G @ X4 ) @ ( F @ I3 ) ) ) )
=> ( ! [J3: $o] :
( ( member_o @ J3 @ B4 )
=> ? [X4: real] :
( ( member_real @ X4 @ A )
& ( ord_less_eq_o @ ( F @ X4 ) @ ( G @ J3 ) ) ) )
=> ( ( complete_Inf_Inf_o @ ( image_real_o @ F @ A ) )
= ( complete_Inf_Inf_o @ ( image_o_o @ G @ B4 ) ) ) ) ) ).
% INF_eq
thf(fact_983_INF__eq,axiom,
! [A: set_nat,B4: set_real,G: real > $o,F: nat > $o] :
( ! [I3: nat] :
( ( member_nat @ I3 @ A )
=> ? [X4: real] :
( ( member_real @ X4 @ B4 )
& ( ord_less_eq_o @ ( G @ X4 ) @ ( F @ I3 ) ) ) )
=> ( ! [J3: real] :
( ( member_real @ J3 @ B4 )
=> ? [X4: nat] :
( ( member_nat @ X4 @ A )
& ( ord_less_eq_o @ ( F @ X4 ) @ ( G @ J3 ) ) ) )
=> ( ( complete_Inf_Inf_o @ ( image_nat_o @ F @ A ) )
= ( complete_Inf_Inf_o @ ( image_real_o @ G @ B4 ) ) ) ) ) ).
% INF_eq
thf(fact_984_INF__eq,axiom,
! [A: set_nat,B4: set_nat,G: nat > $o,F: nat > $o] :
( ! [I3: nat] :
( ( member_nat @ I3 @ A )
=> ? [X4: nat] :
( ( member_nat @ X4 @ B4 )
& ( ord_less_eq_o @ ( G @ X4 ) @ ( F @ I3 ) ) ) )
=> ( ! [J3: nat] :
( ( member_nat @ J3 @ B4 )
=> ? [X4: nat] :
( ( member_nat @ X4 @ A )
& ( ord_less_eq_o @ ( F @ X4 ) @ ( G @ J3 ) ) ) )
=> ( ( complete_Inf_Inf_o @ ( image_nat_o @ F @ A ) )
= ( complete_Inf_Inf_o @ ( image_nat_o @ G @ B4 ) ) ) ) ) ).
% INF_eq
thf(fact_985_INF__eq,axiom,
! [A: set_nat,B4: set_o,G: $o > $o,F: nat > $o] :
( ! [I3: nat] :
( ( member_nat @ I3 @ A )
=> ? [X4: $o] :
( ( member_o @ X4 @ B4 )
& ( ord_less_eq_o @ ( G @ X4 ) @ ( F @ I3 ) ) ) )
=> ( ! [J3: $o] :
( ( member_o @ J3 @ B4 )
=> ? [X4: nat] :
( ( member_nat @ X4 @ A )
& ( ord_less_eq_o @ ( F @ X4 ) @ ( G @ J3 ) ) ) )
=> ( ( complete_Inf_Inf_o @ ( image_nat_o @ F @ A ) )
= ( complete_Inf_Inf_o @ ( image_o_o @ G @ B4 ) ) ) ) ) ).
% INF_eq
thf(fact_986_INF__eq,axiom,
! [A: set_o,B4: set_real,G: real > $o,F: $o > $o] :
( ! [I3: $o] :
( ( member_o @ I3 @ A )
=> ? [X4: real] :
( ( member_real @ X4 @ B4 )
& ( ord_less_eq_o @ ( G @ X4 ) @ ( F @ I3 ) ) ) )
=> ( ! [J3: real] :
( ( member_real @ J3 @ B4 )
=> ? [X4: $o] :
( ( member_o @ X4 @ A )
& ( ord_less_eq_o @ ( F @ X4 ) @ ( G @ J3 ) ) ) )
=> ( ( complete_Inf_Inf_o @ ( image_o_o @ F @ A ) )
= ( complete_Inf_Inf_o @ ( image_real_o @ G @ B4 ) ) ) ) ) ).
% INF_eq
thf(fact_987_INF__eq,axiom,
! [A: set_o,B4: set_nat,G: nat > $o,F: $o > $o] :
( ! [I3: $o] :
( ( member_o @ I3 @ A )
=> ? [X4: nat] :
( ( member_nat @ X4 @ B4 )
& ( ord_less_eq_o @ ( G @ X4 ) @ ( F @ I3 ) ) ) )
=> ( ! [J3: nat] :
( ( member_nat @ J3 @ B4 )
=> ? [X4: $o] :
( ( member_o @ X4 @ A )
& ( ord_less_eq_o @ ( F @ X4 ) @ ( G @ J3 ) ) ) )
=> ( ( complete_Inf_Inf_o @ ( image_o_o @ F @ A ) )
= ( complete_Inf_Inf_o @ ( image_nat_o @ G @ B4 ) ) ) ) ) ).
% INF_eq
thf(fact_988_INF__eq,axiom,
! [A: set_o,B4: set_o,G: $o > $o,F: $o > $o] :
( ! [I3: $o] :
( ( member_o @ I3 @ A )
=> ? [X4: $o] :
( ( member_o @ X4 @ B4 )
& ( ord_less_eq_o @ ( G @ X4 ) @ ( F @ I3 ) ) ) )
=> ( ! [J3: $o] :
( ( member_o @ J3 @ B4 )
=> ? [X4: $o] :
( ( member_o @ X4 @ A )
& ( ord_less_eq_o @ ( F @ X4 ) @ ( G @ J3 ) ) ) )
=> ( ( complete_Inf_Inf_o @ ( image_o_o @ F @ A ) )
= ( complete_Inf_Inf_o @ ( image_o_o @ G @ B4 ) ) ) ) ) ).
% INF_eq
thf(fact_989_INF__eq,axiom,
! [A: set_real,B4: set_real,G: real > set_real,F: real > set_real] :
( ! [I3: real] :
( ( member_real @ I3 @ A )
=> ? [X4: real] :
( ( member_real @ X4 @ B4 )
& ( ord_less_eq_set_real @ ( G @ X4 ) @ ( F @ I3 ) ) ) )
=> ( ! [J3: real] :
( ( member_real @ J3 @ B4 )
=> ? [X4: real] :
( ( member_real @ X4 @ A )
& ( ord_less_eq_set_real @ ( F @ X4 ) @ ( G @ J3 ) ) ) )
=> ( ( comple8289635161444856091t_real @ ( image_real_set_real @ F @ A ) )
= ( comple8289635161444856091t_real @ ( image_real_set_real @ G @ B4 ) ) ) ) ) ).
% INF_eq
thf(fact_990_SUP__eq__const,axiom,
! [I5: set_o,F: $o > set_real,X: set_real] :
( ( I5 != bot_bot_set_o )
=> ( ! [I3: $o] :
( ( member_o @ I3 @ I5 )
=> ( ( F @ I3 )
= X ) )
=> ( ( comple3096694443085538997t_real @ ( image_o_set_real @ F @ I5 ) )
= X ) ) ) ).
% SUP_eq_const
thf(fact_991_SUP__eq__const,axiom,
! [I5: set_set_real,F: set_real > set_real,X: set_real] :
( ( I5 != bot_bot_set_set_real )
=> ( ! [I3: set_real] :
( ( member_set_real @ I3 @ I5 )
=> ( ( F @ I3 )
= X ) )
=> ( ( comple3096694443085538997t_real @ ( image_2436557299294012491t_real @ F @ I5 ) )
= X ) ) ) ).
% SUP_eq_const
thf(fact_992_SUP__eq__const,axiom,
! [I5: set_real,F: real > set_real,X: set_real] :
( ( I5 != bot_bot_set_real )
=> ( ! [I3: real] :
( ( member_real @ I3 @ I5 )
=> ( ( F @ I3 )
= X ) )
=> ( ( comple3096694443085538997t_real @ ( image_real_set_real @ F @ I5 ) )
= X ) ) ) ).
% SUP_eq_const
thf(fact_993_SUP__eq__const,axiom,
! [I5: set_nat,F: nat > set_real,X: set_real] :
( ( I5 != bot_bot_set_nat )
=> ( ! [I3: nat] :
( ( member_nat @ I3 @ I5 )
=> ( ( F @ I3 )
= X ) )
=> ( ( comple3096694443085538997t_real @ ( image_nat_set_real @ F @ I5 ) )
= X ) ) ) ).
% SUP_eq_const
thf(fact_994_SUP__eq__const,axiom,
! [I5: set_o,F: $o > $o,X: $o] :
( ( I5 != bot_bot_set_o )
=> ( ! [I3: $o] :
( ( member_o @ I3 @ I5 )
=> ( ( F @ I3 )
= X ) )
=> ( ( complete_Sup_Sup_o @ ( image_o_o @ F @ I5 ) )
= X ) ) ) ).
% SUP_eq_const
thf(fact_995_SUP__eq__const,axiom,
! [I5: set_set_real,F: set_real > $o,X: $o] :
( ( I5 != bot_bot_set_set_real )
=> ( ! [I3: set_real] :
( ( member_set_real @ I3 @ I5 )
=> ( ( F @ I3 )
= X ) )
=> ( ( complete_Sup_Sup_o @ ( image_set_real_o @ F @ I5 ) )
= X ) ) ) ).
% SUP_eq_const
thf(fact_996_SUP__eq__const,axiom,
! [I5: set_real,F: real > $o,X: $o] :
( ( I5 != bot_bot_set_real )
=> ( ! [I3: real] :
( ( member_real @ I3 @ I5 )
=> ( ( F @ I3 )
= X ) )
=> ( ( complete_Sup_Sup_o @ ( image_real_o @ F @ I5 ) )
= X ) ) ) ).
% SUP_eq_const
thf(fact_997_SUP__eq__const,axiom,
! [I5: set_nat,F: nat > $o,X: $o] :
( ( I5 != bot_bot_set_nat )
=> ( ! [I3: nat] :
( ( member_nat @ I3 @ I5 )
=> ( ( F @ I3 )
= X ) )
=> ( ( complete_Sup_Sup_o @ ( image_nat_o @ F @ I5 ) )
= X ) ) ) ).
% SUP_eq_const
thf(fact_998_ex__less__of__nat__mult,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ? [N3: nat] : ( ord_less_real @ Y @ ( times_times_real @ ( semiri5074537144036343181t_real @ N3 ) @ X ) ) ) ).
% ex_less_of_nat_mult
thf(fact_999_INF__eq__const,axiom,
! [I5: set_o,F: $o > $o,X: $o] :
( ( I5 != bot_bot_set_o )
=> ( ! [I3: $o] :
( ( member_o @ I3 @ I5 )
=> ( ( F @ I3 )
= X ) )
=> ( ( complete_Inf_Inf_o @ ( image_o_o @ F @ I5 ) )
= X ) ) ) ).
% INF_eq_const
thf(fact_1000_INF__eq__const,axiom,
! [I5: set_set_real,F: set_real > $o,X: $o] :
( ( I5 != bot_bot_set_set_real )
=> ( ! [I3: set_real] :
( ( member_set_real @ I3 @ I5 )
=> ( ( F @ I3 )
= X ) )
=> ( ( complete_Inf_Inf_o @ ( image_set_real_o @ F @ I5 ) )
= X ) ) ) ).
% INF_eq_const
thf(fact_1001_INF__eq__const,axiom,
! [I5: set_real,F: real > $o,X: $o] :
( ( I5 != bot_bot_set_real )
=> ( ! [I3: real] :
( ( member_real @ I3 @ I5 )
=> ( ( F @ I3 )
= X ) )
=> ( ( complete_Inf_Inf_o @ ( image_real_o @ F @ I5 ) )
= X ) ) ) ).
% INF_eq_const
thf(fact_1002_INF__eq__const,axiom,
! [I5: set_nat,F: nat > $o,X: $o] :
( ( I5 != bot_bot_set_nat )
=> ( ! [I3: nat] :
( ( member_nat @ I3 @ I5 )
=> ( ( F @ I3 )
= X ) )
=> ( ( complete_Inf_Inf_o @ ( image_nat_o @ F @ I5 ) )
= X ) ) ) ).
% INF_eq_const
thf(fact_1003_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A2: real,B: real,C: real] :
( ( ord_less_real @ A2 @ B )
=> ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( times_times_real @ C @ A2 ) @ ( times_times_real @ C @ B ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_1004_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ C @ A2 ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_1005_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A2: int,B: int,C: int] :
( ( ord_less_int @ A2 @ B )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_1006_mult__less__cancel__right__disj,axiom,
! [A2: real,C: real,B: real] :
( ( ord_less_real @ ( times_times_real @ A2 @ C ) @ ( times_times_real @ B @ C ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
& ( ord_less_real @ A2 @ B ) )
| ( ( ord_less_real @ C @ zero_zero_real )
& ( ord_less_real @ B @ A2 ) ) ) ) ).
% mult_less_cancel_right_disj
thf(fact_1007_mult__less__cancel__right__disj,axiom,
! [A2: int,C: int,B: int] :
( ( ord_less_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B @ C ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
& ( ord_less_int @ A2 @ B ) )
| ( ( ord_less_int @ C @ zero_zero_int )
& ( ord_less_int @ B @ A2 ) ) ) ) ).
% mult_less_cancel_right_disj
thf(fact_1008_mult__strict__right__mono,axiom,
! [A2: real,B: real,C: real] :
( ( ord_less_real @ A2 @ B )
=> ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( times_times_real @ A2 @ C ) @ ( times_times_real @ B @ C ) ) ) ) ).
% mult_strict_right_mono
thf(fact_1009_mult__strict__right__mono,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% mult_strict_right_mono
thf(fact_1010_mult__strict__right__mono,axiom,
! [A2: int,B: int,C: int] :
( ( ord_less_int @ A2 @ B )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% mult_strict_right_mono
thf(fact_1011_mult__strict__right__mono__neg,axiom,
! [B: real,A2: real,C: real] :
( ( ord_less_real @ B @ A2 )
=> ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ A2 @ C ) @ ( times_times_real @ B @ C ) ) ) ) ).
% mult_strict_right_mono_neg
thf(fact_1012_mult__strict__right__mono__neg,axiom,
! [B: int,A2: int,C: int] :
( ( ord_less_int @ B @ A2 )
=> ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% mult_strict_right_mono_neg
thf(fact_1013_mult__less__cancel__left__disj,axiom,
! [C: real,A2: real,B: real] :
( ( ord_less_real @ ( times_times_real @ C @ A2 ) @ ( times_times_real @ C @ B ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
& ( ord_less_real @ A2 @ B ) )
| ( ( ord_less_real @ C @ zero_zero_real )
& ( ord_less_real @ B @ A2 ) ) ) ) ).
% mult_less_cancel_left_disj
thf(fact_1014_mult__less__cancel__left__disj,axiom,
! [C: int,A2: int,B: int] :
( ( ord_less_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
& ( ord_less_int @ A2 @ B ) )
| ( ( ord_less_int @ C @ zero_zero_int )
& ( ord_less_int @ B @ A2 ) ) ) ) ).
% mult_less_cancel_left_disj
thf(fact_1015_mult__strict__left__mono,axiom,
! [A2: real,B: real,C: real] :
( ( ord_less_real @ A2 @ B )
=> ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( times_times_real @ C @ A2 ) @ ( times_times_real @ C @ B ) ) ) ) ).
% mult_strict_left_mono
thf(fact_1016_mult__strict__left__mono,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ C @ A2 ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% mult_strict_left_mono
thf(fact_1017_mult__strict__left__mono,axiom,
! [A2: int,B: int,C: int] :
( ( ord_less_int @ A2 @ B )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B ) ) ) ) ).
% mult_strict_left_mono
thf(fact_1018_mult__strict__left__mono__neg,axiom,
! [B: real,A2: real,C: real] :
( ( ord_less_real @ B @ A2 )
=> ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ C @ A2 ) @ ( times_times_real @ C @ B ) ) ) ) ).
% mult_strict_left_mono_neg
thf(fact_1019_mult__strict__left__mono__neg,axiom,
! [B: int,A2: int,C: int] :
( ( ord_less_int @ B @ A2 )
=> ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B ) ) ) ) ).
% mult_strict_left_mono_neg
thf(fact_1020_mult__less__cancel__left__pos,axiom,
! [C: real,A2: real,B: real] :
( ( ord_less_real @ zero_zero_real @ C )
=> ( ( ord_less_real @ ( times_times_real @ C @ A2 ) @ ( times_times_real @ C @ B ) )
= ( ord_less_real @ A2 @ B ) ) ) ).
% mult_less_cancel_left_pos
thf(fact_1021_mult__less__cancel__left__pos,axiom,
! [C: int,A2: int,B: int] :
( ( ord_less_int @ zero_zero_int @ C )
=> ( ( ord_less_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B ) )
= ( ord_less_int @ A2 @ B ) ) ) ).
% mult_less_cancel_left_pos
thf(fact_1022_mult__less__cancel__left__neg,axiom,
! [C: real,A2: real,B: real] :
( ( ord_less_real @ C @ zero_zero_real )
=> ( ( ord_less_real @ ( times_times_real @ C @ A2 ) @ ( times_times_real @ C @ B ) )
= ( ord_less_real @ B @ A2 ) ) ) ).
% mult_less_cancel_left_neg
thf(fact_1023_mult__less__cancel__left__neg,axiom,
! [C: int,A2: int,B: int] :
( ( ord_less_int @ C @ zero_zero_int )
=> ( ( ord_less_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B ) )
= ( ord_less_int @ B @ A2 ) ) ) ).
% mult_less_cancel_left_neg
thf(fact_1024_zero__less__mult__pos2,axiom,
! [B: real,A2: real] :
( ( ord_less_real @ zero_zero_real @ ( times_times_real @ B @ A2 ) )
=> ( ( ord_less_real @ zero_zero_real @ A2 )
=> ( ord_less_real @ zero_zero_real @ B ) ) ) ).
% zero_less_mult_pos2
thf(fact_1025_zero__less__mult__pos2,axiom,
! [B: nat,A2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ B @ A2 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ A2 )
=> ( ord_less_nat @ zero_zero_nat @ B ) ) ) ).
% zero_less_mult_pos2
thf(fact_1026_zero__less__mult__pos2,axiom,
! [B: int,A2: int] :
( ( ord_less_int @ zero_zero_int @ ( times_times_int @ B @ A2 ) )
=> ( ( ord_less_int @ zero_zero_int @ A2 )
=> ( ord_less_int @ zero_zero_int @ B ) ) ) ).
% zero_less_mult_pos2
thf(fact_1027_zero__less__mult__pos,axiom,
! [A2: real,B: real] :
( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A2 @ B ) )
=> ( ( ord_less_real @ zero_zero_real @ A2 )
=> ( ord_less_real @ zero_zero_real @ B ) ) ) ).
% zero_less_mult_pos
thf(fact_1028_zero__less__mult__pos,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A2 @ B ) )
=> ( ( ord_less_nat @ zero_zero_nat @ A2 )
=> ( ord_less_nat @ zero_zero_nat @ B ) ) ) ).
% zero_less_mult_pos
thf(fact_1029_zero__less__mult__pos,axiom,
! [A2: int,B: int] :
( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A2 @ B ) )
=> ( ( ord_less_int @ zero_zero_int @ A2 )
=> ( ord_less_int @ zero_zero_int @ B ) ) ) ).
% zero_less_mult_pos
thf(fact_1030_zero__less__mult__iff,axiom,
! [A2: real,B: real] :
( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A2 @ B ) )
= ( ( ( ord_less_real @ zero_zero_real @ A2 )
& ( ord_less_real @ zero_zero_real @ B ) )
| ( ( ord_less_real @ A2 @ zero_zero_real )
& ( ord_less_real @ B @ zero_zero_real ) ) ) ) ).
% zero_less_mult_iff
thf(fact_1031_zero__less__mult__iff,axiom,
! [A2: int,B: int] :
( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A2 @ B ) )
= ( ( ( ord_less_int @ zero_zero_int @ A2 )
& ( ord_less_int @ zero_zero_int @ B ) )
| ( ( ord_less_int @ A2 @ zero_zero_int )
& ( ord_less_int @ B @ zero_zero_int ) ) ) ) ).
% zero_less_mult_iff
thf(fact_1032_mult__pos__neg2,axiom,
! [A2: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A2 )
=> ( ( ord_less_real @ B @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ B @ A2 ) @ zero_zero_real ) ) ) ).
% mult_pos_neg2
thf(fact_1033_mult__pos__neg2,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( times_times_nat @ B @ A2 ) @ zero_zero_nat ) ) ) ).
% mult_pos_neg2
thf(fact_1034_mult__pos__neg2,axiom,
! [A2: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A2 )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ B @ A2 ) @ zero_zero_int ) ) ) ).
% mult_pos_neg2
thf(fact_1035_mult__pos__pos,axiom,
! [A2: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A2 )
=> ( ( ord_less_real @ zero_zero_real @ B )
=> ( ord_less_real @ zero_zero_real @ ( times_times_real @ A2 @ B ) ) ) ) ).
% mult_pos_pos
thf(fact_1036_mult__pos__pos,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A2 @ B ) ) ) ) ).
% mult_pos_pos
thf(fact_1037_mult__pos__pos,axiom,
! [A2: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A2 )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ zero_zero_int @ ( times_times_int @ A2 @ B ) ) ) ) ).
% mult_pos_pos
thf(fact_1038_mult__pos__neg,axiom,
! [A2: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A2 )
=> ( ( ord_less_real @ B @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ A2 @ B ) @ zero_zero_real ) ) ) ).
% mult_pos_neg
thf(fact_1039_mult__pos__neg,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( times_times_nat @ A2 @ B ) @ zero_zero_nat ) ) ) ).
% mult_pos_neg
thf(fact_1040_mult__pos__neg,axiom,
! [A2: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A2 )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ A2 @ B ) @ zero_zero_int ) ) ) ).
% mult_pos_neg
thf(fact_1041_mult__neg__pos,axiom,
! [A2: real,B: real] :
( ( ord_less_real @ A2 @ zero_zero_real )
=> ( ( ord_less_real @ zero_zero_real @ B )
=> ( ord_less_real @ ( times_times_real @ A2 @ B ) @ zero_zero_real ) ) ) ).
% mult_neg_pos
thf(fact_1042_mult__neg__pos,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ zero_zero_nat )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ ( times_times_nat @ A2 @ B ) @ zero_zero_nat ) ) ) ).
% mult_neg_pos
thf(fact_1043_mult__neg__pos,axiom,
! [A2: int,B: int] :
( ( ord_less_int @ A2 @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ ( times_times_int @ A2 @ B ) @ zero_zero_int ) ) ) ).
% mult_neg_pos
thf(fact_1044_mult__less__0__iff,axiom,
! [A2: real,B: real] :
( ( ord_less_real @ ( times_times_real @ A2 @ B ) @ zero_zero_real )
= ( ( ( ord_less_real @ zero_zero_real @ A2 )
& ( ord_less_real @ B @ zero_zero_real ) )
| ( ( ord_less_real @ A2 @ zero_zero_real )
& ( ord_less_real @ zero_zero_real @ B ) ) ) ) ).
% mult_less_0_iff
thf(fact_1045_mult__less__0__iff,axiom,
! [A2: int,B: int] :
( ( ord_less_int @ ( times_times_int @ A2 @ B ) @ zero_zero_int )
= ( ( ( ord_less_int @ zero_zero_int @ A2 )
& ( ord_less_int @ B @ zero_zero_int ) )
| ( ( ord_less_int @ A2 @ zero_zero_int )
& ( ord_less_int @ zero_zero_int @ B ) ) ) ) ).
% mult_less_0_iff
thf(fact_1046_not__square__less__zero,axiom,
! [A2: real] :
~ ( ord_less_real @ ( times_times_real @ A2 @ A2 ) @ zero_zero_real ) ).
% not_square_less_zero
thf(fact_1047_not__square__less__zero,axiom,
! [A2: int] :
~ ( ord_less_int @ ( times_times_int @ A2 @ A2 ) @ zero_zero_int ) ).
% not_square_less_zero
thf(fact_1048_mult__neg__neg,axiom,
! [A2: real,B: real] :
( ( ord_less_real @ A2 @ zero_zero_real )
=> ( ( ord_less_real @ B @ zero_zero_real )
=> ( ord_less_real @ zero_zero_real @ ( times_times_real @ A2 @ B ) ) ) ) ).
% mult_neg_neg
thf(fact_1049_mult__neg__neg,axiom,
! [A2: int,B: int] :
( ( ord_less_int @ A2 @ zero_zero_int )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ zero_zero_int @ ( times_times_int @ A2 @ B ) ) ) ) ).
% mult_neg_neg
thf(fact_1050_less__iff__diff__less__0,axiom,
( ord_less_real
= ( ^ [A3: real,B2: real] : ( ord_less_real @ ( minus_minus_real @ A3 @ B2 ) @ zero_zero_real ) ) ) ).
% less_iff_diff_less_0
thf(fact_1051_less__iff__diff__less__0,axiom,
( ord_less_int
= ( ^ [A3: int,B2: int] : ( ord_less_int @ ( minus_minus_int @ A3 @ B2 ) @ zero_zero_int ) ) ) ).
% less_iff_diff_less_0
thf(fact_1052_divide__strict__right__mono__neg,axiom,
! [B: real,A2: real,C: real] :
( ( ord_less_real @ B @ A2 )
=> ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ ( divide_divide_real @ A2 @ C ) @ ( divide_divide_real @ B @ C ) ) ) ) ).
% divide_strict_right_mono_neg
thf(fact_1053_divide__strict__right__mono,axiom,
! [A2: real,B: real,C: real] :
( ( ord_less_real @ A2 @ B )
=> ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( divide_divide_real @ A2 @ C ) @ ( divide_divide_real @ B @ C ) ) ) ) ).
% divide_strict_right_mono
thf(fact_1054_zero__less__divide__iff,axiom,
! [A2: real,B: real] :
( ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ A2 @ B ) )
= ( ( ( ord_less_real @ zero_zero_real @ A2 )
& ( ord_less_real @ zero_zero_real @ B ) )
| ( ( ord_less_real @ A2 @ zero_zero_real )
& ( ord_less_real @ B @ zero_zero_real ) ) ) ) ).
% zero_less_divide_iff
thf(fact_1055_divide__less__cancel,axiom,
! [A2: real,C: real,B: real] :
( ( ord_less_real @ ( divide_divide_real @ A2 @ C ) @ ( divide_divide_real @ B @ C ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ A2 @ B ) )
& ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ B @ A2 ) )
& ( C != zero_zero_real ) ) ) ).
% divide_less_cancel
thf(fact_1056_divide__less__0__iff,axiom,
! [A2: real,B: real] :
( ( ord_less_real @ ( divide_divide_real @ A2 @ B ) @ zero_zero_real )
= ( ( ( ord_less_real @ zero_zero_real @ A2 )
& ( ord_less_real @ B @ zero_zero_real ) )
| ( ( ord_less_real @ A2 @ zero_zero_real )
& ( ord_less_real @ zero_zero_real @ B ) ) ) ) ).
% divide_less_0_iff
thf(fact_1057_divide__pos__pos,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ zero_zero_real @ Y )
=> ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ X @ Y ) ) ) ) ).
% divide_pos_pos
thf(fact_1058_divide__pos__neg,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ Y @ zero_zero_real )
=> ( ord_less_real @ ( divide_divide_real @ X @ Y ) @ zero_zero_real ) ) ) ).
% divide_pos_neg
thf(fact_1059_divide__neg__pos,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ zero_zero_real )
=> ( ( ord_less_real @ zero_zero_real @ Y )
=> ( ord_less_real @ ( divide_divide_real @ X @ Y ) @ zero_zero_real ) ) ) ).
% divide_neg_pos
thf(fact_1060_divide__neg__neg,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ zero_zero_real )
=> ( ( ord_less_real @ Y @ zero_zero_real )
=> ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ X @ Y ) ) ) ) ).
% divide_neg_neg
thf(fact_1061_of__nat__less__0__iff,axiom,
! [M: nat] :
~ ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ zero_zero_nat ) ).
% of_nat_less_0_iff
thf(fact_1062_of__nat__less__0__iff,axiom,
! [M: nat] :
~ ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ zero_zero_real ) ).
% of_nat_less_0_iff
thf(fact_1063_of__nat__less__0__iff,axiom,
! [M: nat] :
~ ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ zero_zero_int ) ).
% of_nat_less_0_iff
thf(fact_1064_abs__not__less__zero,axiom,
! [A2: real] :
~ ( ord_less_real @ ( abs_abs_real @ A2 ) @ zero_zero_real ) ).
% abs_not_less_zero
thf(fact_1065_abs__not__less__zero,axiom,
! [A2: int] :
~ ( ord_less_int @ ( abs_abs_int @ A2 ) @ zero_zero_int ) ).
% abs_not_less_zero
thf(fact_1066_abs__of__pos,axiom,
! [A2: real] :
( ( ord_less_real @ zero_zero_real @ A2 )
=> ( ( abs_abs_real @ A2 )
= A2 ) ) ).
% abs_of_pos
thf(fact_1067_abs__of__pos,axiom,
! [A2: int] :
( ( ord_less_int @ zero_zero_int @ A2 )
=> ( ( abs_abs_int @ A2 )
= A2 ) ) ).
% abs_of_pos
thf(fact_1068_less__cSupD,axiom,
! [X6: set_int,Z2: int] :
( ( X6 != bot_bot_set_int )
=> ( ( ord_less_int @ Z2 @ ( complete_Sup_Sup_int @ X6 ) )
=> ? [X3: int] :
( ( member_int @ X3 @ X6 )
& ( ord_less_int @ Z2 @ X3 ) ) ) ) ).
% less_cSupD
thf(fact_1069_less__cSupD,axiom,
! [X6: set_real,Z2: real] :
( ( X6 != bot_bot_set_real )
=> ( ( ord_less_real @ Z2 @ ( comple1385675409528146559p_real @ X6 ) )
=> ? [X3: real] :
( ( member_real @ X3 @ X6 )
& ( ord_less_real @ Z2 @ X3 ) ) ) ) ).
% less_cSupD
thf(fact_1070_less__cSupD,axiom,
! [X6: set_nat,Z2: nat] :
( ( X6 != bot_bot_set_nat )
=> ( ( ord_less_nat @ Z2 @ ( complete_Sup_Sup_nat @ X6 ) )
=> ? [X3: nat] :
( ( member_nat @ X3 @ X6 )
& ( ord_less_nat @ Z2 @ X3 ) ) ) ) ).
% less_cSupD
thf(fact_1071_less__cSupE,axiom,
! [Y: int,X6: set_int] :
( ( ord_less_int @ Y @ ( complete_Sup_Sup_int @ X6 ) )
=> ( ( X6 != bot_bot_set_int )
=> ~ ! [X3: int] :
( ( member_int @ X3 @ X6 )
=> ~ ( ord_less_int @ Y @ X3 ) ) ) ) ).
% less_cSupE
thf(fact_1072_less__cSupE,axiom,
! [Y: real,X6: set_real] :
( ( ord_less_real @ Y @ ( comple1385675409528146559p_real @ X6 ) )
=> ( ( X6 != bot_bot_set_real )
=> ~ ! [X3: real] :
( ( member_real @ X3 @ X6 )
=> ~ ( ord_less_real @ Y @ X3 ) ) ) ) ).
% less_cSupE
thf(fact_1073_less__cSupE,axiom,
! [Y: nat,X6: set_nat] :
( ( ord_less_nat @ Y @ ( complete_Sup_Sup_nat @ X6 ) )
=> ( ( X6 != bot_bot_set_nat )
=> ~ ! [X3: nat] :
( ( member_nat @ X3 @ X6 )
=> ~ ( ord_less_nat @ Y @ X3 ) ) ) ) ).
% less_cSupE
thf(fact_1074_abs__mult__less,axiom,
! [A2: real,C: real,B: real,D: real] :
( ( ord_less_real @ ( abs_abs_real @ A2 ) @ C )
=> ( ( ord_less_real @ ( abs_abs_real @ B ) @ D )
=> ( ord_less_real @ ( times_times_real @ ( abs_abs_real @ A2 ) @ ( abs_abs_real @ B ) ) @ ( times_times_real @ C @ D ) ) ) ) ).
% abs_mult_less
thf(fact_1075_abs__mult__less,axiom,
! [A2: int,C: int,B: int,D: int] :
( ( ord_less_int @ ( abs_abs_int @ A2 ) @ C )
=> ( ( ord_less_int @ ( abs_abs_int @ B ) @ D )
=> ( ord_less_int @ ( times_times_int @ ( abs_abs_int @ A2 ) @ ( abs_abs_int @ B ) ) @ ( times_times_int @ C @ D ) ) ) ) ).
% abs_mult_less
thf(fact_1076_cInf__lessD,axiom,
! [X6: set_int,Z2: int] :
( ( X6 != bot_bot_set_int )
=> ( ( ord_less_int @ ( complete_Inf_Inf_int @ X6 ) @ Z2 )
=> ? [X3: int] :
( ( member_int @ X3 @ X6 )
& ( ord_less_int @ X3 @ Z2 ) ) ) ) ).
% cInf_lessD
thf(fact_1077_cInf__lessD,axiom,
! [X6: set_real,Z2: real] :
( ( X6 != bot_bot_set_real )
=> ( ( ord_less_real @ ( comple4887499456419720421f_real @ X6 ) @ Z2 )
=> ? [X3: real] :
( ( member_real @ X3 @ X6 )
& ( ord_less_real @ X3 @ Z2 ) ) ) ) ).
% cInf_lessD
thf(fact_1078_cInf__lessD,axiom,
! [X6: set_nat,Z2: nat] :
( ( X6 != bot_bot_set_nat )
=> ( ( ord_less_nat @ ( complete_Inf_Inf_nat @ X6 ) @ Z2 )
=> ? [X3: nat] :
( ( member_nat @ X3 @ X6 )
& ( ord_less_nat @ X3 @ Z2 ) ) ) ) ).
% cInf_lessD
thf(fact_1079_reals__Archimedean3,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ! [Y4: real] :
? [N3: nat] : ( ord_less_real @ Y4 @ ( times_times_real @ ( semiri5074537144036343181t_real @ N3 ) @ X ) ) ) ).
% reals_Archimedean3
thf(fact_1080_image__mult__atLeastAtMost__if,axiom,
! [C: real,X: real,Y: real] :
( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ( image_real_real @ ( times_times_real @ C ) @ ( set_or1222579329274155063t_real @ X @ Y ) )
= ( set_or1222579329274155063t_real @ ( times_times_real @ C @ X ) @ ( times_times_real @ C @ Y ) ) ) )
& ( ~ ( ord_less_real @ zero_zero_real @ C )
=> ( ( ( ord_less_eq_real @ X @ Y )
=> ( ( image_real_real @ ( times_times_real @ C ) @ ( set_or1222579329274155063t_real @ X @ Y ) )
= ( set_or1222579329274155063t_real @ ( times_times_real @ C @ Y ) @ ( times_times_real @ C @ X ) ) ) )
& ( ~ ( ord_less_eq_real @ X @ Y )
=> ( ( image_real_real @ ( times_times_real @ C ) @ ( set_or1222579329274155063t_real @ X @ Y ) )
= bot_bot_set_real ) ) ) ) ) ).
% image_mult_atLeastAtMost_if
thf(fact_1081_SUP__upper2,axiom,
! [I2: set_real,A: set_set_real,U2: set_real,F: set_real > set_real] :
( ( member_set_real @ I2 @ A )
=> ( ( ord_less_eq_set_real @ U2 @ ( F @ I2 ) )
=> ( ord_less_eq_set_real @ U2 @ ( comple3096694443085538997t_real @ ( image_2436557299294012491t_real @ F @ A ) ) ) ) ) ).
% SUP_upper2
thf(fact_1082_SUP__upper2,axiom,
! [I2: real,A: set_real,U2: set_real,F: real > set_real] :
( ( member_real @ I2 @ A )
=> ( ( ord_less_eq_set_real @ U2 @ ( F @ I2 ) )
=> ( ord_less_eq_set_real @ U2 @ ( comple3096694443085538997t_real @ ( image_real_set_real @ F @ A ) ) ) ) ) ).
% SUP_upper2
thf(fact_1083_SUP__upper2,axiom,
! [I2: nat,A: set_nat,U2: set_real,F: nat > set_real] :
( ( member_nat @ I2 @ A )
=> ( ( ord_less_eq_set_real @ U2 @ ( F @ I2 ) )
=> ( ord_less_eq_set_real @ U2 @ ( comple3096694443085538997t_real @ ( image_nat_set_real @ F @ A ) ) ) ) ) ).
% SUP_upper2
thf(fact_1084_SUP__upper2,axiom,
! [I2: $o,A: set_o,U2: set_real,F: $o > set_real] :
( ( member_o @ I2 @ A )
=> ( ( ord_less_eq_set_real @ U2 @ ( F @ I2 ) )
=> ( ord_less_eq_set_real @ U2 @ ( comple3096694443085538997t_real @ ( image_o_set_real @ F @ A ) ) ) ) ) ).
% SUP_upper2
thf(fact_1085_SUP__upper2,axiom,
! [I2: set_real,A: set_set_real,U2: $o,F: set_real > $o] :
( ( member_set_real @ I2 @ A )
=> ( ( ord_less_eq_o @ U2 @ ( F @ I2 ) )
=> ( ord_less_eq_o @ U2 @ ( complete_Sup_Sup_o @ ( image_set_real_o @ F @ A ) ) ) ) ) ).
% SUP_upper2
thf(fact_1086_SUP__upper2,axiom,
! [I2: real,A: set_real,U2: $o,F: real > $o] :
( ( member_real @ I2 @ A )
=> ( ( ord_less_eq_o @ U2 @ ( F @ I2 ) )
=> ( ord_less_eq_o @ U2 @ ( complete_Sup_Sup_o @ ( image_real_o @ F @ A ) ) ) ) ) ).
% SUP_upper2
thf(fact_1087_SUP__upper2,axiom,
! [I2: nat,A: set_nat,U2: $o,F: nat > $o] :
( ( member_nat @ I2 @ A )
=> ( ( ord_less_eq_o @ U2 @ ( F @ I2 ) )
=> ( ord_less_eq_o @ U2 @ ( complete_Sup_Sup_o @ ( image_nat_o @ F @ A ) ) ) ) ) ).
% SUP_upper2
thf(fact_1088_SUP__upper2,axiom,
! [I2: $o,A: set_o,U2: $o,F: $o > $o] :
( ( member_o @ I2 @ A )
=> ( ( ord_less_eq_o @ U2 @ ( F @ I2 ) )
=> ( ord_less_eq_o @ U2 @ ( complete_Sup_Sup_o @ ( image_o_o @ F @ A ) ) ) ) ) ).
% SUP_upper2
thf(fact_1089_SUP__upper,axiom,
! [I2: set_real,A: set_set_real,F: set_real > set_real] :
( ( member_set_real @ I2 @ A )
=> ( ord_less_eq_set_real @ ( F @ I2 ) @ ( comple3096694443085538997t_real @ ( image_2436557299294012491t_real @ F @ A ) ) ) ) ).
% SUP_upper
thf(fact_1090_SUP__upper,axiom,
! [I2: real,A: set_real,F: real > set_real] :
( ( member_real @ I2 @ A )
=> ( ord_less_eq_set_real @ ( F @ I2 ) @ ( comple3096694443085538997t_real @ ( image_real_set_real @ F @ A ) ) ) ) ).
% SUP_upper
thf(fact_1091_SUP__upper,axiom,
! [I2: nat,A: set_nat,F: nat > set_real] :
( ( member_nat @ I2 @ A )
=> ( ord_less_eq_set_real @ ( F @ I2 ) @ ( comple3096694443085538997t_real @ ( image_nat_set_real @ F @ A ) ) ) ) ).
% SUP_upper
thf(fact_1092_SUP__upper,axiom,
! [I2: $o,A: set_o,F: $o > set_real] :
( ( member_o @ I2 @ A )
=> ( ord_less_eq_set_real @ ( F @ I2 ) @ ( comple3096694443085538997t_real @ ( image_o_set_real @ F @ A ) ) ) ) ).
% SUP_upper
thf(fact_1093_SUP__upper,axiom,
! [I2: set_real,A: set_set_real,F: set_real > $o] :
( ( member_set_real @ I2 @ A )
=> ( ord_less_eq_o @ ( F @ I2 ) @ ( complete_Sup_Sup_o @ ( image_set_real_o @ F @ A ) ) ) ) ).
% SUP_upper
thf(fact_1094_SUP__upper,axiom,
! [I2: real,A: set_real,F: real > $o] :
( ( member_real @ I2 @ A )
=> ( ord_less_eq_o @ ( F @ I2 ) @ ( complete_Sup_Sup_o @ ( image_real_o @ F @ A ) ) ) ) ).
% SUP_upper
thf(fact_1095_SUP__upper,axiom,
! [I2: nat,A: set_nat,F: nat > $o] :
( ( member_nat @ I2 @ A )
=> ( ord_less_eq_o @ ( F @ I2 ) @ ( complete_Sup_Sup_o @ ( image_nat_o @ F @ A ) ) ) ) ).
% SUP_upper
thf(fact_1096_SUP__upper,axiom,
! [I2: $o,A: set_o,F: $o > $o] :
( ( member_o @ I2 @ A )
=> ( ord_less_eq_o @ ( F @ I2 ) @ ( complete_Sup_Sup_o @ ( image_o_o @ F @ A ) ) ) ) ).
% SUP_upper
thf(fact_1097_SUP__least,axiom,
! [A: set_set_real,F: set_real > set_real,U2: set_real] :
( ! [I3: set_real] :
( ( member_set_real @ I3 @ A )
=> ( ord_less_eq_set_real @ ( F @ I3 ) @ U2 ) )
=> ( ord_less_eq_set_real @ ( comple3096694443085538997t_real @ ( image_2436557299294012491t_real @ F @ A ) ) @ U2 ) ) ).
% SUP_least
thf(fact_1098_SUP__least,axiom,
! [A: set_real,F: real > set_real,U2: set_real] :
( ! [I3: real] :
( ( member_real @ I3 @ A )
=> ( ord_less_eq_set_real @ ( F @ I3 ) @ U2 ) )
=> ( ord_less_eq_set_real @ ( comple3096694443085538997t_real @ ( image_real_set_real @ F @ A ) ) @ U2 ) ) ).
% SUP_least
thf(fact_1099_SUP__least,axiom,
! [A: set_nat,F: nat > set_real,U2: set_real] :
( ! [I3: nat] :
( ( member_nat @ I3 @ A )
=> ( ord_less_eq_set_real @ ( F @ I3 ) @ U2 ) )
=> ( ord_less_eq_set_real @ ( comple3096694443085538997t_real @ ( image_nat_set_real @ F @ A ) ) @ U2 ) ) ).
% SUP_least
thf(fact_1100_SUP__least,axiom,
! [A: set_o,F: $o > set_real,U2: set_real] :
( ! [I3: $o] :
( ( member_o @ I3 @ A )
=> ( ord_less_eq_set_real @ ( F @ I3 ) @ U2 ) )
=> ( ord_less_eq_set_real @ ( comple3096694443085538997t_real @ ( image_o_set_real @ F @ A ) ) @ U2 ) ) ).
% SUP_least
thf(fact_1101_SUP__least,axiom,
! [A: set_set_real,F: set_real > $o,U2: $o] :
( ! [I3: set_real] :
( ( member_set_real @ I3 @ A )
=> ( ord_less_eq_o @ ( F @ I3 ) @ U2 ) )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ ( image_set_real_o @ F @ A ) ) @ U2 ) ) ).
% SUP_least
thf(fact_1102_SUP__least,axiom,
! [A: set_real,F: real > $o,U2: $o] :
( ! [I3: real] :
( ( member_real @ I3 @ A )
=> ( ord_less_eq_o @ ( F @ I3 ) @ U2 ) )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ ( image_real_o @ F @ A ) ) @ U2 ) ) ).
% SUP_least
thf(fact_1103_SUP__least,axiom,
! [A: set_nat,F: nat > $o,U2: $o] :
( ! [I3: nat] :
( ( member_nat @ I3 @ A )
=> ( ord_less_eq_o @ ( F @ I3 ) @ U2 ) )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ ( image_nat_o @ F @ A ) ) @ U2 ) ) ).
% SUP_least
thf(fact_1104_SUP__least,axiom,
! [A: set_o,F: $o > $o,U2: $o] :
( ! [I3: $o] :
( ( member_o @ I3 @ A )
=> ( ord_less_eq_o @ ( F @ I3 ) @ U2 ) )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ ( image_o_o @ F @ A ) ) @ U2 ) ) ).
% SUP_least
thf(fact_1105_SUP__eqI,axiom,
! [A: set_set_real,F: set_real > set_real,X: set_real] :
( ! [I3: set_real] :
( ( member_set_real @ I3 @ A )
=> ( ord_less_eq_set_real @ ( F @ I3 ) @ X ) )
=> ( ! [Y2: set_real] :
( ! [I4: set_real] :
( ( member_set_real @ I4 @ A )
=> ( ord_less_eq_set_real @ ( F @ I4 ) @ Y2 ) )
=> ( ord_less_eq_set_real @ X @ Y2 ) )
=> ( ( comple3096694443085538997t_real @ ( image_2436557299294012491t_real @ F @ A ) )
= X ) ) ) ).
% SUP_eqI
thf(fact_1106_SUP__eqI,axiom,
! [A: set_real,F: real > set_real,X: set_real] :
( ! [I3: real] :
( ( member_real @ I3 @ A )
=> ( ord_less_eq_set_real @ ( F @ I3 ) @ X ) )
=> ( ! [Y2: set_real] :
( ! [I4: real] :
( ( member_real @ I4 @ A )
=> ( ord_less_eq_set_real @ ( F @ I4 ) @ Y2 ) )
=> ( ord_less_eq_set_real @ X @ Y2 ) )
=> ( ( comple3096694443085538997t_real @ ( image_real_set_real @ F @ A ) )
= X ) ) ) ).
% SUP_eqI
thf(fact_1107_SUP__eqI,axiom,
! [A: set_nat,F: nat > set_real,X: set_real] :
( ! [I3: nat] :
( ( member_nat @ I3 @ A )
=> ( ord_less_eq_set_real @ ( F @ I3 ) @ X ) )
=> ( ! [Y2: set_real] :
( ! [I4: nat] :
( ( member_nat @ I4 @ A )
=> ( ord_less_eq_set_real @ ( F @ I4 ) @ Y2 ) )
=> ( ord_less_eq_set_real @ X @ Y2 ) )
=> ( ( comple3096694443085538997t_real @ ( image_nat_set_real @ F @ A ) )
= X ) ) ) ).
% SUP_eqI
thf(fact_1108_SUP__eqI,axiom,
! [A: set_o,F: $o > set_real,X: set_real] :
( ! [I3: $o] :
( ( member_o @ I3 @ A )
=> ( ord_less_eq_set_real @ ( F @ I3 ) @ X ) )
=> ( ! [Y2: set_real] :
( ! [I4: $o] :
( ( member_o @ I4 @ A )
=> ( ord_less_eq_set_real @ ( F @ I4 ) @ Y2 ) )
=> ( ord_less_eq_set_real @ X @ Y2 ) )
=> ( ( comple3096694443085538997t_real @ ( image_o_set_real @ F @ A ) )
= X ) ) ) ).
% SUP_eqI
thf(fact_1109_SUP__eqI,axiom,
! [A: set_set_real,F: set_real > $o,X: $o] :
( ! [I3: set_real] :
( ( member_set_real @ I3 @ A )
=> ( ord_less_eq_o @ ( F @ I3 ) @ X ) )
=> ( ! [Y2: $o] :
( ! [I4: set_real] :
( ( member_set_real @ I4 @ A )
=> ( ord_less_eq_o @ ( F @ I4 ) @ Y2 ) )
=> ( ord_less_eq_o @ X @ Y2 ) )
=> ( ( complete_Sup_Sup_o @ ( image_set_real_o @ F @ A ) )
= X ) ) ) ).
% SUP_eqI
thf(fact_1110_SUP__eqI,axiom,
! [A: set_real,F: real > $o,X: $o] :
( ! [I3: real] :
( ( member_real @ I3 @ A )
=> ( ord_less_eq_o @ ( F @ I3 ) @ X ) )
=> ( ! [Y2: $o] :
( ! [I4: real] :
( ( member_real @ I4 @ A )
=> ( ord_less_eq_o @ ( F @ I4 ) @ Y2 ) )
=> ( ord_less_eq_o @ X @ Y2 ) )
=> ( ( complete_Sup_Sup_o @ ( image_real_o @ F @ A ) )
= X ) ) ) ).
% SUP_eqI
thf(fact_1111_SUP__eqI,axiom,
! [A: set_nat,F: nat > $o,X: $o] :
( ! [I3: nat] :
( ( member_nat @ I3 @ A )
=> ( ord_less_eq_o @ ( F @ I3 ) @ X ) )
=> ( ! [Y2: $o] :
( ! [I4: nat] :
( ( member_nat @ I4 @ A )
=> ( ord_less_eq_o @ ( F @ I4 ) @ Y2 ) )
=> ( ord_less_eq_o @ X @ Y2 ) )
=> ( ( complete_Sup_Sup_o @ ( image_nat_o @ F @ A ) )
= X ) ) ) ).
% SUP_eqI
thf(fact_1112_SUP__eqI,axiom,
! [A: set_o,F: $o > $o,X: $o] :
( ! [I3: $o] :
( ( member_o @ I3 @ A )
=> ( ord_less_eq_o @ ( F @ I3 ) @ X ) )
=> ( ! [Y2: $o] :
( ! [I4: $o] :
( ( member_o @ I4 @ A )
=> ( ord_less_eq_o @ ( F @ I4 ) @ Y2 ) )
=> ( ord_less_eq_o @ X @ Y2 ) )
=> ( ( complete_Sup_Sup_o @ ( image_o_o @ F @ A ) )
= X ) ) ) ).
% SUP_eqI
thf(fact_1113_INF__greatest,axiom,
! [A: set_set_real,U2: set_real,F: set_real > set_real] :
( ! [I3: set_real] :
( ( member_set_real @ I3 @ A )
=> ( ord_less_eq_set_real @ U2 @ ( F @ I3 ) ) )
=> ( ord_less_eq_set_real @ U2 @ ( comple8289635161444856091t_real @ ( image_2436557299294012491t_real @ F @ A ) ) ) ) ).
% INF_greatest
thf(fact_1114_INF__greatest,axiom,
! [A: set_real,U2: set_real,F: real > set_real] :
( ! [I3: real] :
( ( member_real @ I3 @ A )
=> ( ord_less_eq_set_real @ U2 @ ( F @ I3 ) ) )
=> ( ord_less_eq_set_real @ U2 @ ( comple8289635161444856091t_real @ ( image_real_set_real @ F @ A ) ) ) ) ).
% INF_greatest
thf(fact_1115_INF__greatest,axiom,
! [A: set_nat,U2: set_real,F: nat > set_real] :
( ! [I3: nat] :
( ( member_nat @ I3 @ A )
=> ( ord_less_eq_set_real @ U2 @ ( F @ I3 ) ) )
=> ( ord_less_eq_set_real @ U2 @ ( comple8289635161444856091t_real @ ( image_nat_set_real @ F @ A ) ) ) ) ).
% INF_greatest
thf(fact_1116_INF__greatest,axiom,
! [A: set_o,U2: set_real,F: $o > set_real] :
( ! [I3: $o] :
( ( member_o @ I3 @ A )
=> ( ord_less_eq_set_real @ U2 @ ( F @ I3 ) ) )
=> ( ord_less_eq_set_real @ U2 @ ( comple8289635161444856091t_real @ ( image_o_set_real @ F @ A ) ) ) ) ).
% INF_greatest
thf(fact_1117_INF__greatest,axiom,
! [A: set_set_real,U2: $o,F: set_real > $o] :
( ! [I3: set_real] :
( ( member_set_real @ I3 @ A )
=> ( ord_less_eq_o @ U2 @ ( F @ I3 ) ) )
=> ( ord_less_eq_o @ U2 @ ( complete_Inf_Inf_o @ ( image_set_real_o @ F @ A ) ) ) ) ).
% INF_greatest
thf(fact_1118_INF__greatest,axiom,
! [A: set_real,U2: $o,F: real > $o] :
( ! [I3: real] :
( ( member_real @ I3 @ A )
=> ( ord_less_eq_o @ U2 @ ( F @ I3 ) ) )
=> ( ord_less_eq_o @ U2 @ ( complete_Inf_Inf_o @ ( image_real_o @ F @ A ) ) ) ) ).
% INF_greatest
thf(fact_1119_INF__greatest,axiom,
! [A: set_nat,U2: $o,F: nat > $o] :
( ! [I3: nat] :
( ( member_nat @ I3 @ A )
=> ( ord_less_eq_o @ U2 @ ( F @ I3 ) ) )
=> ( ord_less_eq_o @ U2 @ ( complete_Inf_Inf_o @ ( image_nat_o @ F @ A ) ) ) ) ).
% INF_greatest
thf(fact_1120_INF__greatest,axiom,
! [A: set_o,U2: $o,F: $o > $o] :
( ! [I3: $o] :
( ( member_o @ I3 @ A )
=> ( ord_less_eq_o @ U2 @ ( F @ I3 ) ) )
=> ( ord_less_eq_o @ U2 @ ( complete_Inf_Inf_o @ ( image_o_o @ F @ A ) ) ) ) ).
% INF_greatest
thf(fact_1121_INF__lower2,axiom,
! [I2: set_real,A: set_set_real,F: set_real > set_real,U2: set_real] :
( ( member_set_real @ I2 @ A )
=> ( ( ord_less_eq_set_real @ ( F @ I2 ) @ U2 )
=> ( ord_less_eq_set_real @ ( comple8289635161444856091t_real @ ( image_2436557299294012491t_real @ F @ A ) ) @ U2 ) ) ) ).
% INF_lower2
thf(fact_1122_INF__lower2,axiom,
! [I2: real,A: set_real,F: real > set_real,U2: set_real] :
( ( member_real @ I2 @ A )
=> ( ( ord_less_eq_set_real @ ( F @ I2 ) @ U2 )
=> ( ord_less_eq_set_real @ ( comple8289635161444856091t_real @ ( image_real_set_real @ F @ A ) ) @ U2 ) ) ) ).
% INF_lower2
thf(fact_1123_INF__lower2,axiom,
! [I2: nat,A: set_nat,F: nat > set_real,U2: set_real] :
( ( member_nat @ I2 @ A )
=> ( ( ord_less_eq_set_real @ ( F @ I2 ) @ U2 )
=> ( ord_less_eq_set_real @ ( comple8289635161444856091t_real @ ( image_nat_set_real @ F @ A ) ) @ U2 ) ) ) ).
% INF_lower2
thf(fact_1124_INF__lower2,axiom,
! [I2: $o,A: set_o,F: $o > set_real,U2: set_real] :
( ( member_o @ I2 @ A )
=> ( ( ord_less_eq_set_real @ ( F @ I2 ) @ U2 )
=> ( ord_less_eq_set_real @ ( comple8289635161444856091t_real @ ( image_o_set_real @ F @ A ) ) @ U2 ) ) ) ).
% INF_lower2
thf(fact_1125_INF__lower2,axiom,
! [I2: set_real,A: set_set_real,F: set_real > $o,U2: $o] :
( ( member_set_real @ I2 @ A )
=> ( ( ord_less_eq_o @ ( F @ I2 ) @ U2 )
=> ( ord_less_eq_o @ ( complete_Inf_Inf_o @ ( image_set_real_o @ F @ A ) ) @ U2 ) ) ) ).
% INF_lower2
thf(fact_1126_INF__lower2,axiom,
! [I2: real,A: set_real,F: real > $o,U2: $o] :
( ( member_real @ I2 @ A )
=> ( ( ord_less_eq_o @ ( F @ I2 ) @ U2 )
=> ( ord_less_eq_o @ ( complete_Inf_Inf_o @ ( image_real_o @ F @ A ) ) @ U2 ) ) ) ).
% INF_lower2
thf(fact_1127_INF__lower2,axiom,
! [I2: nat,A: set_nat,F: nat > $o,U2: $o] :
( ( member_nat @ I2 @ A )
=> ( ( ord_less_eq_o @ ( F @ I2 ) @ U2 )
=> ( ord_less_eq_o @ ( complete_Inf_Inf_o @ ( image_nat_o @ F @ A ) ) @ U2 ) ) ) ).
% INF_lower2
thf(fact_1128_INF__lower2,axiom,
! [I2: $o,A: set_o,F: $o > $o,U2: $o] :
( ( member_o @ I2 @ A )
=> ( ( ord_less_eq_o @ ( F @ I2 ) @ U2 )
=> ( ord_less_eq_o @ ( complete_Inf_Inf_o @ ( image_o_o @ F @ A ) ) @ U2 ) ) ) ).
% INF_lower2
thf(fact_1129_INF__lower,axiom,
! [I2: set_real,A: set_set_real,F: set_real > set_real] :
( ( member_set_real @ I2 @ A )
=> ( ord_less_eq_set_real @ ( comple8289635161444856091t_real @ ( image_2436557299294012491t_real @ F @ A ) ) @ ( F @ I2 ) ) ) ).
% INF_lower
thf(fact_1130_INF__lower,axiom,
! [I2: real,A: set_real,F: real > set_real] :
( ( member_real @ I2 @ A )
=> ( ord_less_eq_set_real @ ( comple8289635161444856091t_real @ ( image_real_set_real @ F @ A ) ) @ ( F @ I2 ) ) ) ).
% INF_lower
thf(fact_1131_INF__lower,axiom,
! [I2: nat,A: set_nat,F: nat > set_real] :
( ( member_nat @ I2 @ A )
=> ( ord_less_eq_set_real @ ( comple8289635161444856091t_real @ ( image_nat_set_real @ F @ A ) ) @ ( F @ I2 ) ) ) ).
% INF_lower
thf(fact_1132_INF__lower,axiom,
! [I2: $o,A: set_o,F: $o > set_real] :
( ( member_o @ I2 @ A )
=> ( ord_less_eq_set_real @ ( comple8289635161444856091t_real @ ( image_o_set_real @ F @ A ) ) @ ( F @ I2 ) ) ) ).
% INF_lower
thf(fact_1133_INF__lower,axiom,
! [I2: set_real,A: set_set_real,F: set_real > $o] :
( ( member_set_real @ I2 @ A )
=> ( ord_less_eq_o @ ( complete_Inf_Inf_o @ ( image_set_real_o @ F @ A ) ) @ ( F @ I2 ) ) ) ).
% INF_lower
thf(fact_1134_INF__lower,axiom,
! [I2: real,A: set_real,F: real > $o] :
( ( member_real @ I2 @ A )
=> ( ord_less_eq_o @ ( complete_Inf_Inf_o @ ( image_real_o @ F @ A ) ) @ ( F @ I2 ) ) ) ).
% INF_lower
thf(fact_1135_INF__lower,axiom,
! [I2: nat,A: set_nat,F: nat > $o] :
( ( member_nat @ I2 @ A )
=> ( ord_less_eq_o @ ( complete_Inf_Inf_o @ ( image_nat_o @ F @ A ) ) @ ( F @ I2 ) ) ) ).
% INF_lower
thf(fact_1136_INF__lower,axiom,
! [I2: $o,A: set_o,F: $o > $o] :
( ( member_o @ I2 @ A )
=> ( ord_less_eq_o @ ( complete_Inf_Inf_o @ ( image_o_o @ F @ A ) ) @ ( F @ I2 ) ) ) ).
% INF_lower
thf(fact_1137_INF__eqI,axiom,
! [A: set_set_real,X: set_real,F: set_real > set_real] :
( ! [I3: set_real] :
( ( member_set_real @ I3 @ A )
=> ( ord_less_eq_set_real @ X @ ( F @ I3 ) ) )
=> ( ! [Y2: set_real] :
( ! [I4: set_real] :
( ( member_set_real @ I4 @ A )
=> ( ord_less_eq_set_real @ Y2 @ ( F @ I4 ) ) )
=> ( ord_less_eq_set_real @ Y2 @ X ) )
=> ( ( comple8289635161444856091t_real @ ( image_2436557299294012491t_real @ F @ A ) )
= X ) ) ) ).
% INF_eqI
thf(fact_1138_INF__eqI,axiom,
! [A: set_real,X: set_real,F: real > set_real] :
( ! [I3: real] :
( ( member_real @ I3 @ A )
=> ( ord_less_eq_set_real @ X @ ( F @ I3 ) ) )
=> ( ! [Y2: set_real] :
( ! [I4: real] :
( ( member_real @ I4 @ A )
=> ( ord_less_eq_set_real @ Y2 @ ( F @ I4 ) ) )
=> ( ord_less_eq_set_real @ Y2 @ X ) )
=> ( ( comple8289635161444856091t_real @ ( image_real_set_real @ F @ A ) )
= X ) ) ) ).
% INF_eqI
thf(fact_1139_INF__eqI,axiom,
! [A: set_nat,X: set_real,F: nat > set_real] :
( ! [I3: nat] :
( ( member_nat @ I3 @ A )
=> ( ord_less_eq_set_real @ X @ ( F @ I3 ) ) )
=> ( ! [Y2: set_real] :
( ! [I4: nat] :
( ( member_nat @ I4 @ A )
=> ( ord_less_eq_set_real @ Y2 @ ( F @ I4 ) ) )
=> ( ord_less_eq_set_real @ Y2 @ X ) )
=> ( ( comple8289635161444856091t_real @ ( image_nat_set_real @ F @ A ) )
= X ) ) ) ).
% INF_eqI
thf(fact_1140_INF__eqI,axiom,
! [A: set_o,X: set_real,F: $o > set_real] :
( ! [I3: $o] :
( ( member_o @ I3 @ A )
=> ( ord_less_eq_set_real @ X @ ( F @ I3 ) ) )
=> ( ! [Y2: set_real] :
( ! [I4: $o] :
( ( member_o @ I4 @ A )
=> ( ord_less_eq_set_real @ Y2 @ ( F @ I4 ) ) )
=> ( ord_less_eq_set_real @ Y2 @ X ) )
=> ( ( comple8289635161444856091t_real @ ( image_o_set_real @ F @ A ) )
= X ) ) ) ).
% INF_eqI
thf(fact_1141_INF__eqI,axiom,
! [A: set_set_real,X: $o,F: set_real > $o] :
( ! [I3: set_real] :
( ( member_set_real @ I3 @ A )
=> ( ord_less_eq_o @ X @ ( F @ I3 ) ) )
=> ( ! [Y2: $o] :
( ! [I4: set_real] :
( ( member_set_real @ I4 @ A )
=> ( ord_less_eq_o @ Y2 @ ( F @ I4 ) ) )
=> ( ord_less_eq_o @ Y2 @ X ) )
=> ( ( complete_Inf_Inf_o @ ( image_set_real_o @ F @ A ) )
= X ) ) ) ).
% INF_eqI
thf(fact_1142_INF__eqI,axiom,
! [A: set_real,X: $o,F: real > $o] :
( ! [I3: real] :
( ( member_real @ I3 @ A )
=> ( ord_less_eq_o @ X @ ( F @ I3 ) ) )
=> ( ! [Y2: $o] :
( ! [I4: real] :
( ( member_real @ I4 @ A )
=> ( ord_less_eq_o @ Y2 @ ( F @ I4 ) ) )
=> ( ord_less_eq_o @ Y2 @ X ) )
=> ( ( complete_Inf_Inf_o @ ( image_real_o @ F @ A ) )
= X ) ) ) ).
% INF_eqI
thf(fact_1143_INF__eqI,axiom,
! [A: set_nat,X: $o,F: nat > $o] :
( ! [I3: nat] :
( ( member_nat @ I3 @ A )
=> ( ord_less_eq_o @ X @ ( F @ I3 ) ) )
=> ( ! [Y2: $o] :
( ! [I4: nat] :
( ( member_nat @ I4 @ A )
=> ( ord_less_eq_o @ Y2 @ ( F @ I4 ) ) )
=> ( ord_less_eq_o @ Y2 @ X ) )
=> ( ( complete_Inf_Inf_o @ ( image_nat_o @ F @ A ) )
= X ) ) ) ).
% INF_eqI
thf(fact_1144_INF__eqI,axiom,
! [A: set_o,X: $o,F: $o > $o] :
( ! [I3: $o] :
( ( member_o @ I3 @ A )
=> ( ord_less_eq_o @ X @ ( F @ I3 ) ) )
=> ( ! [Y2: $o] :
( ! [I4: $o] :
( ( member_o @ I4 @ A )
=> ( ord_less_eq_o @ Y2 @ ( F @ I4 ) ) )
=> ( ord_less_eq_o @ Y2 @ X ) )
=> ( ( complete_Inf_Inf_o @ ( image_o_o @ F @ A ) )
= X ) ) ) ).
% INF_eqI
thf(fact_1145_image__mult__atLeastAtMost__if_H,axiom,
! [X: real,Y: real,C: real] :
( ( ( ord_less_eq_real @ X @ Y )
=> ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ( image_real_real
@ ^ [X2: real] : ( times_times_real @ X2 @ C )
@ ( set_or1222579329274155063t_real @ X @ Y ) )
= ( set_or1222579329274155063t_real @ ( times_times_real @ X @ C ) @ ( times_times_real @ Y @ C ) ) ) )
& ( ~ ( ord_less_real @ zero_zero_real @ C )
=> ( ( image_real_real
@ ^ [X2: real] : ( times_times_real @ X2 @ C )
@ ( set_or1222579329274155063t_real @ X @ Y ) )
= ( set_or1222579329274155063t_real @ ( times_times_real @ Y @ C ) @ ( times_times_real @ X @ C ) ) ) ) ) )
& ( ~ ( ord_less_eq_real @ X @ Y )
=> ( ( image_real_real
@ ^ [X2: real] : ( times_times_real @ X2 @ C )
@ ( set_or1222579329274155063t_real @ X @ Y ) )
= bot_bot_set_real ) ) ) ).
% image_mult_atLeastAtMost_if'
thf(fact_1146_SUP__eq__iff,axiom,
! [I5: set_real,C: $o,F: real > $o] :
( ( I5 != bot_bot_set_real )
=> ( ! [I3: real] :
( ( member_real @ I3 @ I5 )
=> ( ord_less_eq_o @ C @ ( F @ I3 ) ) )
=> ( ( ( complete_Sup_Sup_o @ ( image_real_o @ F @ I5 ) )
= C )
= ( ! [X2: real] :
( ( member_real @ X2 @ I5 )
=> ( ( F @ X2 )
= C ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_1147_SUP__eq__iff,axiom,
! [I5: set_nat,C: $o,F: nat > $o] :
( ( I5 != bot_bot_set_nat )
=> ( ! [I3: nat] :
( ( member_nat @ I3 @ I5 )
=> ( ord_less_eq_o @ C @ ( F @ I3 ) ) )
=> ( ( ( complete_Sup_Sup_o @ ( image_nat_o @ F @ I5 ) )
= C )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ I5 )
=> ( ( F @ X2 )
= C ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_1148_complete__real,axiom,
! [S: set_real] :
( ? [X4: real] : ( member_real @ X4 @ S )
=> ( ? [Z3: real] :
! [X3: real] :
( ( member_real @ X3 @ S )
=> ( ord_less_eq_real @ X3 @ Z3 ) )
=> ? [Y2: real] :
( ! [X4: real] :
( ( member_real @ X4 @ S )
=> ( ord_less_eq_real @ X4 @ Y2 ) )
& ! [Z3: real] :
( ! [X3: real] :
( ( member_real @ X3 @ S )
=> ( ord_less_eq_real @ X3 @ Z3 ) )
=> ( ord_less_eq_real @ Y2 @ Z3 ) ) ) ) ) ).
% complete_real
thf(fact_1149_real__of__nat__div4,axiom,
! [N: nat,X: nat] : ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N @ X ) ) @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ X ) ) ) ).
% real_of_nat_div4
thf(fact_1150_square__continuous,axiom,
! [E2: real,X: real] :
( ( ord_less_real @ zero_zero_real @ E2 )
=> ? [D5: real] :
( ( ord_less_real @ zero_zero_real @ D5 )
& ! [Y4: real] :
( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ Y4 @ X ) ) @ D5 )
=> ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ ( times_times_real @ Y4 @ Y4 ) @ ( times_times_real @ X @ X ) ) ) @ E2 ) ) ) ) ).
% square_continuous
thf(fact_1151_lemma__interval,axiom,
! [A2: real,X: real,B: real] :
( ( ord_less_real @ A2 @ X )
=> ( ( ord_less_real @ X @ B )
=> ? [D5: real] :
( ( ord_less_real @ zero_zero_real @ D5 )
& ! [Y4: real] :
( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ Y4 ) ) @ D5 )
=> ( ( ord_less_eq_real @ A2 @ Y4 )
& ( ord_less_eq_real @ Y4 @ B ) ) ) ) ) ) ).
% lemma_interval
thf(fact_1152__092_060open_0620_A_060_An_092_060close_062,axiom,
ord_less_nat @ zero_zero_nat @ n ).
% \<open>0 < n\<close>
thf(fact_1153_bot__nat__0_Onot__eq__extremum,axiom,
! [A2: nat] :
( ( A2 != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ A2 ) ) ).
% bot_nat_0.not_eq_extremum
thf(fact_1154_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_1155_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_1156_an__less__del,axiom,
ord_less_real @ ( divide_divide_real @ a @ ( semiri5074537144036343181t_real @ n ) ) @ ( del @ ( divide_divide_real @ epsilon @ a ) ) ).
% an_less_del
thf(fact_1157_zero__less__diff,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N @ M ) )
= ( ord_less_nat @ M @ N ) ) ).
% zero_less_diff
thf(fact_1158_mult__less__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M @ N ) ) ) ).
% mult_less_cancel2
thf(fact_1159_nat__0__less__mult__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ M @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% nat_0_less_mult_iff
thf(fact_1160_div__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( divide_divide_nat @ M @ N )
= zero_zero_nat ) ) ).
% div_less
thf(fact_1161_div__neg__neg__trivial,axiom,
! [K: int,L: int] :
( ( ord_less_eq_int @ K @ zero_zero_int )
=> ( ( ord_less_int @ L @ K )
=> ( ( divide_divide_int @ K @ L )
= zero_zero_int ) ) ) ).
% div_neg_neg_trivial
thf(fact_1162_div__pos__pos__trivial,axiom,
! [K: int,L: int] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ( ( ord_less_int @ K @ L )
=> ( ( divide_divide_int @ K @ L )
= zero_zero_int ) ) ) ).
% div_pos_pos_trivial
thf(fact_1163_mult__le__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% mult_le_cancel2
thf(fact_1164_div__mult__self__is__m,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( divide_divide_nat @ ( times_times_nat @ M @ N ) @ N )
= M ) ) ).
% div_mult_self_is_m
thf(fact_1165_div__mult__self1__is__m,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( divide_divide_nat @ ( times_times_nat @ N @ M ) @ N )
= M ) ) ).
% div_mult_self1_is_m
thf(fact_1166_linorder__neqE__nat,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neqE_nat
thf(fact_1167_infinite__descent,axiom,
! [P: nat > $o,N: nat] :
( ! [N3: nat] :
( ~ ( P @ N3 )
=> ? [M4: nat] :
( ( ord_less_nat @ M4 @ N3 )
& ~ ( P @ M4 ) ) )
=> ( P @ N ) ) ).
% infinite_descent
thf(fact_1168_nat__less__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N3: nat] :
( ! [M4: nat] :
( ( ord_less_nat @ M4 @ N3 )
=> ( P @ M4 ) )
=> ( P @ N3 ) )
=> ( P @ N ) ) ).
% nat_less_induct
thf(fact_1169_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_1170_less__not__refl3,axiom,
! [S2: nat,T2: nat] :
( ( ord_less_nat @ S2 @ T2 )
=> ( S2 != T2 ) ) ).
% less_not_refl3
thf(fact_1171_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( M != N ) ) ).
% less_not_refl2
thf(fact_1172_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_1173_nat__neq__iff,axiom,
! [M: nat,N: nat] :
( ( M != N )
= ( ( ord_less_nat @ M @ N )
| ( ord_less_nat @ N @ M ) ) ) ).
% nat_neq_iff
thf(fact_1174_image__int__atLeastAtMost,axiom,
! [A2: nat,B: nat] :
( ( image_nat_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ A2 @ B ) )
= ( set_or1266510415728281911st_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).
% image_int_atLeastAtMost
thf(fact_1175_bot__nat__0_Oextremum__strict,axiom,
! [A2: nat] :
~ ( ord_less_nat @ A2 @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_1176_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_1177_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_1178_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_1179_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_1180_gr__implies__not0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_1181_infinite__descent0,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ~ ( P @ N3 )
=> ? [M4: nat] :
( ( ord_less_nat @ M4 @ N3 )
& ~ ( P @ M4 ) ) ) )
=> ( P @ N ) ) ) ).
% infinite_descent0
thf(fact_1182_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I2: nat,J: nat] :
( ! [I3: nat,J3: nat] :
( ( ord_less_nat @ I3 @ J3 )
=> ( ord_less_nat @ ( F @ I3 ) @ ( F @ J3 ) ) )
=> ( ( ord_less_eq_nat @ I2 @ J )
=> ( ord_less_eq_nat @ ( F @ I2 ) @ ( F @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_1183_le__neq__implies__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( M != N )
=> ( ord_less_nat @ M @ N ) ) ) ).
% le_neq_implies_less
thf(fact_1184_less__or__eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( ( ord_less_nat @ M @ N )
| ( M = N ) )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% less_or_eq_imp_le
thf(fact_1185_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M5: nat,N2: nat] :
( ( ord_less_nat @ M5 @ N2 )
| ( M5 = N2 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_1186_less__imp__le__nat,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% less_imp_le_nat
thf(fact_1187_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M5: nat,N2: nat] :
( ( ord_less_eq_nat @ M5 @ N2 )
& ( M5 != N2 ) ) ) ) ).
% nat_less_le
thf(fact_1188_less__int__code_I1_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_int_code(1)
thf(fact_1189_diff__less__mono2,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( ord_less_nat @ M @ L )
=> ( ord_less_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ) ).
% diff_less_mono2
thf(fact_1190_less__imp__diff__less,axiom,
! [J: nat,K: nat,N: nat] :
( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( minus_minus_nat @ J @ N ) @ K ) ) ).
% less_imp_diff_less
thf(fact_1191_zero__less__imp__eq__int,axiom,
! [K: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ? [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
& ( K
= ( semiri1314217659103216013at_int @ N3 ) ) ) ) ).
% zero_less_imp_eq_int
thf(fact_1192_pos__int__cases,axiom,
! [K: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ~ ! [N3: nat] :
( ( K
= ( semiri1314217659103216013at_int @ N3 ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N3 ) ) ) ).
% pos_int_cases
thf(fact_1193_zmult__zless__mono2__lemma,axiom,
! [I2: int,J: int,K: nat] :
( ( ord_less_int @ I2 @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ K ) @ I2 ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ K ) @ J ) ) ) ) ).
% zmult_zless_mono2_lemma
thf(fact_1194_ex__least__nat__le,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K4: nat] :
( ( ord_less_eq_nat @ K4 @ N )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ K4 )
=> ~ ( P @ I4 ) )
& ( P @ K4 ) ) ) ) ).
% ex_least_nat_le
thf(fact_1195_diff__less,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ M ) ) ) ).
% diff_less
thf(fact_1196_mult__less__mono1,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_nat @ ( times_times_nat @ I2 @ K ) @ ( times_times_nat @ J @ K ) ) ) ) ).
% mult_less_mono1
thf(fact_1197_mult__less__mono2,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_nat @ ( times_times_nat @ K @ I2 ) @ ( times_times_nat @ K @ J ) ) ) ) ).
% mult_less_mono2
thf(fact_1198_diff__less__mono,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ C @ A2 )
=> ( ord_less_nat @ ( minus_minus_nat @ A2 @ C ) @ ( minus_minus_nat @ B @ C ) ) ) ) ).
% diff_less_mono
thf(fact_1199_less__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_nat @ M @ N ) ) ) ) ).
% less_diff_iff
thf(fact_1200_zmult__zless__mono2,axiom,
! [I2: int,J: int,K: int] :
( ( ord_less_int @ I2 @ J )
=> ( ( ord_less_int @ zero_zero_int @ K )
=> ( ord_less_int @ ( times_times_int @ K @ I2 ) @ ( times_times_int @ K @ J ) ) ) ) ).
% zmult_zless_mono2
thf(fact_1201_Euclidean__Division_Odiv__eq__0__iff,axiom,
! [M: nat,N: nat] :
( ( ( divide_divide_nat @ M @ N )
= zero_zero_nat )
= ( ( ord_less_nat @ M @ N )
| ( N = zero_zero_nat ) ) ) ).
% Euclidean_Division.div_eq_0_iff
thf(fact_1202_pos__imp__zdiv__neg__iff,axiom,
! [B: int,A2: int] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_int @ ( divide_divide_int @ A2 @ B ) @ zero_zero_int )
= ( ord_less_int @ A2 @ zero_zero_int ) ) ) ).
% pos_imp_zdiv_neg_iff
thf(fact_1203_neg__imp__zdiv__neg__iff,axiom,
! [B: int,A2: int] :
( ( ord_less_int @ B @ zero_zero_int )
=> ( ( ord_less_int @ ( divide_divide_int @ A2 @ B ) @ zero_zero_int )
= ( ord_less_int @ zero_zero_int @ A2 ) ) ) ).
% neg_imp_zdiv_neg_iff
thf(fact_1204_div__neg__pos__less0,axiom,
! [A2: int,B: int] :
( ( ord_less_int @ A2 @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ ( divide_divide_int @ A2 @ B ) @ zero_zero_int ) ) ) ).
% div_neg_pos_less0
thf(fact_1205_less__mult__imp__div__less,axiom,
! [M: nat,I2: nat,N: nat] :
( ( ord_less_nat @ M @ ( times_times_nat @ I2 @ N ) )
=> ( ord_less_nat @ ( divide_divide_nat @ M @ N ) @ I2 ) ) ).
% less_mult_imp_div_less
thf(fact_1206_div__le__mono2,axiom,
! [M: nat,N: nat,K: nat] :
( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( divide_divide_nat @ K @ N ) @ ( divide_divide_nat @ K @ M ) ) ) ) ).
% div_le_mono2
thf(fact_1207_div__greater__zero__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ M @ N ) )
= ( ( ord_less_eq_nat @ N @ M )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% div_greater_zero_iff
thf(fact_1208_div__less__iff__less__mult,axiom,
! [Q: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ Q )
=> ( ( ord_less_nat @ ( divide_divide_nat @ M @ Q ) @ N )
= ( ord_less_nat @ M @ ( times_times_nat @ N @ Q ) ) ) ) ).
% div_less_iff_less_mult
thf(fact_1209_zdiv__mono1,axiom,
! [A2: int,A5: int,B: int] :
( ( ord_less_eq_int @ A2 @ A5 )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ ( divide_divide_int @ A2 @ B ) @ ( divide_divide_int @ A5 @ B ) ) ) ) ).
% zdiv_mono1
thf(fact_1210_zdiv__mono2,axiom,
! [A2: int,B5: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A2 )
=> ( ( ord_less_int @ zero_zero_int @ B5 )
=> ( ( ord_less_eq_int @ B5 @ B )
=> ( ord_less_eq_int @ ( divide_divide_int @ A2 @ B ) @ ( divide_divide_int @ A2 @ B5 ) ) ) ) ) ).
% zdiv_mono2
thf(fact_1211_zdiv__eq__0__iff,axiom,
! [I2: int,K: int] :
( ( ( divide_divide_int @ I2 @ K )
= zero_zero_int )
= ( ( K = zero_zero_int )
| ( ( ord_less_eq_int @ zero_zero_int @ I2 )
& ( ord_less_int @ I2 @ K ) )
| ( ( ord_less_eq_int @ I2 @ zero_zero_int )
& ( ord_less_int @ K @ I2 ) ) ) ) ).
% zdiv_eq_0_iff
thf(fact_1212_zdiv__mono1__neg,axiom,
! [A2: int,A5: int,B: int] :
( ( ord_less_eq_int @ A2 @ A5 )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ ( divide_divide_int @ A5 @ B ) @ ( divide_divide_int @ A2 @ B ) ) ) ) ).
% zdiv_mono1_neg
thf(fact_1213_zdiv__mono2__neg,axiom,
! [A2: int,B5: int,B: int] :
( ( ord_less_int @ A2 @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B5 )
=> ( ( ord_less_eq_int @ B5 @ B )
=> ( ord_less_eq_int @ ( divide_divide_int @ A2 @ B5 ) @ ( divide_divide_int @ A2 @ B ) ) ) ) ) ).
% zdiv_mono2_neg
thf(fact_1214_div__int__pos__iff,axiom,
! [K: int,L: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ K @ L ) )
= ( ( K = zero_zero_int )
| ( L = zero_zero_int )
| ( ( ord_less_eq_int @ zero_zero_int @ K )
& ( ord_less_eq_int @ zero_zero_int @ L ) )
| ( ( ord_less_int @ K @ zero_zero_int )
& ( ord_less_int @ L @ zero_zero_int ) ) ) ) ).
% div_int_pos_iff
thf(fact_1215_div__nonneg__neg__le0,axiom,
! [A2: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A2 )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ ( divide_divide_int @ A2 @ B ) @ zero_zero_int ) ) ) ).
% div_nonneg_neg_le0
thf(fact_1216_div__nonpos__pos__le0,axiom,
! [A2: int,B: int] :
( ( ord_less_eq_int @ A2 @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ ( divide_divide_int @ A2 @ B ) @ zero_zero_int ) ) ) ).
% div_nonpos_pos_le0
thf(fact_1217_pos__imp__zdiv__pos__iff,axiom,
! [K: int,I2: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ( ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ I2 @ K ) )
= ( ord_less_eq_int @ K @ I2 ) ) ) ).
% pos_imp_zdiv_pos_iff
thf(fact_1218_neg__imp__zdiv__nonneg__iff,axiom,
! [B: int,A2: int] :
( ( ord_less_int @ B @ zero_zero_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ A2 @ B ) )
= ( ord_less_eq_int @ A2 @ zero_zero_int ) ) ) ).
% neg_imp_zdiv_nonneg_iff
thf(fact_1219_pos__imp__zdiv__nonneg__iff,axiom,
! [B: int,A2: int] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ A2 @ B ) )
= ( ord_less_eq_int @ zero_zero_int @ A2 ) ) ) ).
% pos_imp_zdiv_nonneg_iff
thf(fact_1220_nonneg1__imp__zdiv__pos__iff,axiom,
! [A2: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A2 )
=> ( ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ A2 @ B ) )
= ( ( ord_less_eq_int @ B @ A2 )
& ( ord_less_int @ zero_zero_int @ B ) ) ) ) ).
% nonneg1_imp_zdiv_pos_iff
thf(fact_1221_less__eq__div__iff__mult__less__eq,axiom,
! [Q: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ Q )
=> ( ( ord_less_eq_nat @ M @ ( divide_divide_nat @ N @ Q ) )
= ( ord_less_eq_nat @ ( times_times_nat @ M @ Q ) @ N ) ) ) ).
% less_eq_div_iff_mult_less_eq
thf(fact_1222_real__archimedian__rdiv__eq__0,axiom,
! [X: real,C: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ! [M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ M2 )
=> ( ord_less_eq_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ M2 ) @ X ) @ C ) )
=> ( X = zero_zero_real ) ) ) ) ).
% real_archimedian_rdiv_eq_0
thf(fact_1223_lemma__interval__lt,axiom,
! [A2: real,X: real,B: real] :
( ( ord_less_real @ A2 @ X )
=> ( ( ord_less_real @ X @ B )
=> ? [D5: real] :
( ( ord_less_real @ zero_zero_real @ D5 )
& ! [Y4: real] :
( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ Y4 ) ) @ D5 )
=> ( ( ord_less_real @ A2 @ Y4 )
& ( ord_less_real @ Y4 @ B ) ) ) ) ) ) ).
% lemma_interval_lt
thf(fact_1224_nat__mult__le__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% nat_mult_le_cancel_disj
thf(fact_1225_nat__mult__less__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M @ N ) ) ) ).
% nat_mult_less_cancel_disj
thf(fact_1226_zdiff__int__split,axiom,
! [P: int > $o,X: nat,Y: nat] :
( ( P @ ( semiri1314217659103216013at_int @ ( minus_minus_nat @ X @ Y ) ) )
= ( ( ( ord_less_eq_nat @ Y @ X )
=> ( P @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ X ) @ ( semiri1314217659103216013at_int @ Y ) ) ) )
& ( ( ord_less_nat @ X @ Y )
=> ( P @ zero_zero_int ) ) ) ) ).
% zdiff_int_split
thf(fact_1227_Inf__bool__def,axiom,
( complete_Inf_Inf_o
= ( ^ [A6: set_o] :
~ ( member_o @ $false @ A6 ) ) ) ).
% Inf_bool_def
thf(fact_1228_Sup__bool__def,axiom,
( complete_Sup_Sup_o
= ( member_o @ $true ) ) ).
% Sup_bool_def
thf(fact_1229_nat__mult__eq__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N ) )
= ( ( K = zero_zero_nat )
| ( M = N ) ) ) ).
% nat_mult_eq_cancel_disj
thf(fact_1230_conj__le__cong,axiom,
! [X: int,X8: int,P: $o,P2: $o] :
( ( X = X8 )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X8 )
=> ( P = P2 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X )
& P )
= ( ( ord_less_eq_int @ zero_zero_int @ X8 )
& P2 ) ) ) ) ).
% conj_le_cong
thf(fact_1231_imp__le__cong,axiom,
! [X: int,X8: int,P: $o,P2: $o] :
( ( X = X8 )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X8 )
=> ( P = P2 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X )
=> P )
= ( ( ord_less_eq_int @ zero_zero_int @ X8 )
=> P2 ) ) ) ) ).
% imp_le_cong
thf(fact_1232_nat__mult__less__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ord_less_nat @ M @ N ) ) ) ).
% nat_mult_less_cancel1
thf(fact_1233_nat__mult__eq__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N ) )
= ( M = N ) ) ) ).
% nat_mult_eq_cancel1
thf(fact_1234_nat__mult__div__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( K = zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= zero_zero_nat ) )
& ( ( K != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( divide_divide_nat @ M @ N ) ) ) ) ).
% nat_mult_div_cancel_disj
thf(fact_1235_minusinfinity,axiom,
! [D: int,P1: int > $o,P: int > $o] :
( ( ord_less_int @ zero_zero_int @ D )
=> ( ! [X3: int,K4: int] :
( ( P1 @ X3 )
= ( P1 @ ( minus_minus_int @ X3 @ ( times_times_int @ K4 @ D ) ) ) )
=> ( ? [Z3: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z3 )
=> ( ( P @ X3 )
= ( P1 @ X3 ) ) )
=> ( ? [X_12: int] : ( P1 @ X_12 )
=> ? [X_1: int] : ( P @ X_1 ) ) ) ) ) ).
% minusinfinity
thf(fact_1236_plusinfinity,axiom,
! [D: int,P2: int > $o,P: int > $o] :
( ( ord_less_int @ zero_zero_int @ D )
=> ( ! [X3: int,K4: int] :
( ( P2 @ X3 )
= ( P2 @ ( minus_minus_int @ X3 @ ( times_times_int @ K4 @ D ) ) ) )
=> ( ? [Z3: int] :
! [X3: int] :
( ( ord_less_int @ Z3 @ X3 )
=> ( ( P @ X3 )
= ( P2 @ X3 ) ) )
=> ( ? [X_12: int] : ( P2 @ X_12 )
=> ? [X_1: int] : ( P @ X_1 ) ) ) ) ) ).
% plusinfinity
thf(fact_1237_nat__mult__le__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ) ).
% nat_mult_le_cancel1
thf(fact_1238_nat__mult__div__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( divide_divide_nat @ M @ N ) ) ) ).
% nat_mult_div_cancel1
thf(fact_1239_decr__mult__lemma,axiom,
! [D: int,P: int > $o,K: int] :
( ( ord_less_int @ zero_zero_int @ D )
=> ( ! [X3: int] :
( ( P @ X3 )
=> ( P @ ( minus_minus_int @ X3 @ D ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ K )
=> ! [X4: int] :
( ( P @ X4 )
=> ( P @ ( minus_minus_int @ X4 @ ( times_times_int @ K @ D ) ) ) ) ) ) ) ).
% decr_mult_lemma
thf(fact_1240_int__ops_I6_J,axiom,
! [A2: nat,B: nat] :
( ( ( ord_less_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B ) )
=> ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ A2 @ B ) )
= zero_zero_int ) )
& ( ~ ( ord_less_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B ) )
=> ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ A2 @ B ) )
= ( minus_minus_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B ) ) ) ) ) ).
% int_ops(6)
thf(fact_1241_verit__la__generic,axiom,
! [A2: int,X: int] :
( ( ord_less_eq_int @ A2 @ X )
| ( A2 = X )
| ( ord_less_eq_int @ X @ A2 ) ) ).
% verit_la_generic
thf(fact_1242_nat__int__comparison_I1_J,axiom,
( ( ^ [Y3: nat,Z: nat] : ( Y3 = Z ) )
= ( ^ [A3: nat,B2: nat] :
( ( semiri1314217659103216013at_int @ A3 )
= ( semiri1314217659103216013at_int @ B2 ) ) ) ) ).
% nat_int_comparison(1)
thf(fact_1243_int__if,axiom,
! [P: $o,A2: nat,B: nat] :
( ( P
=> ( ( semiri1314217659103216013at_int @ ( if_nat @ P @ A2 @ B ) )
= ( semiri1314217659103216013at_int @ A2 ) ) )
& ( ~ P
=> ( ( semiri1314217659103216013at_int @ ( if_nat @ P @ A2 @ B ) )
= ( semiri1314217659103216013at_int @ B ) ) ) ) ).
% int_if
thf(fact_1244_int__ops_I1_J,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% int_ops(1)
thf(fact_1245_nat__int__comparison_I3_J,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B2: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ).
% nat_int_comparison(3)
thf(fact_1246_nat__int__comparison_I2_J,axiom,
( ord_less_nat
= ( ^ [A3: nat,B2: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ).
% nat_int_comparison(2)
thf(fact_1247_int__ops_I7_J,axiom,
! [A2: nat,B: nat] :
( ( semiri1314217659103216013at_int @ ( times_times_nat @ A2 @ B ) )
= ( times_times_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).
% int_ops(7)
thf(fact_1248_int__ops_I8_J,axiom,
! [A2: nat,B: nat] :
( ( semiri1314217659103216013at_int @ ( divide_divide_nat @ A2 @ B ) )
= ( divide_divide_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).
% int_ops(8)
thf(fact_1249_nat__leq__as__int,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B2: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ).
% nat_leq_as_int
thf(fact_1250_nat__less__as__int,axiom,
( ord_less_nat
= ( ^ [A3: nat,B2: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ).
% nat_less_as_int
thf(fact_1251__092_060open_062continuous__on_A_1230_O_Ob_125_Ag_092_060close_062,axiom,
topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ zero_zero_real @ b ) @ g ).
% \<open>continuous_on {0..b} g\<close>
thf(fact_1252_f1__def,axiom,
( f1
= ( comp_real_real_real @ f @ lower ) ) ).
% f1_def
thf(fact_1253_f2__def,axiom,
( f2
= ( comp_real_real_real @ f @ upper ) ) ).
% f2_def
thf(fact_1254_cont__0a,axiom,
topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ zero_zero_real @ a ) @ f ).
% cont_0a
thf(fact_1255_cont,axiom,
topolo5044208981011980120l_real @ ( set_ord_atLeast_real @ zero_zero_real ) @ f ).
% cont
thf(fact_1256_continuous__image__closed__interval,axiom,
! [A2: real,B: real,F: real > real] :
( ( ord_less_eq_real @ A2 @ B )
=> ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A2 @ B ) @ F )
=> ? [C5: real,D5: real] :
( ( ( image_real_real @ F @ ( set_or1222579329274155063t_real @ A2 @ B ) )
= ( set_or1222579329274155063t_real @ C5 @ D5 ) )
& ( ord_less_eq_real @ C5 @ D5 ) ) ) ) ).
% continuous_image_closed_interval
thf(fact_1257_atLeastAtMost__subset__contains__Inf,axiom,
! [A: set_real,A2: real,B: real] :
( ( A != bot_bot_set_real )
=> ( ( ord_less_eq_real @ A2 @ B )
=> ( ( ord_less_eq_set_real @ A @ ( set_or1222579329274155063t_real @ A2 @ B ) )
=> ( member_real @ ( comple4887499456419720421f_real @ A ) @ ( set_or1222579329274155063t_real @ A2 @ B ) ) ) ) ) ).
% atLeastAtMost_subset_contains_Inf
thf(fact_1258_sm,axiom,
monoto4017252874604999745l_real @ ( set_ord_atLeast_real @ zero_zero_real ) @ ord_less_real @ ord_less_real @ f ).
% sm
thf(fact_1259_Bolzano,axiom,
! [A2: real,B: real,P: real > real > $o] :
( ( ord_less_eq_real @ A2 @ B )
=> ( ! [A4: real,B3: real,C5: real] :
( ( P @ A4 @ B3 )
=> ( ( P @ B3 @ C5 )
=> ( ( ord_less_eq_real @ A4 @ B3 )
=> ( ( ord_less_eq_real @ B3 @ C5 )
=> ( P @ A4 @ C5 ) ) ) ) )
=> ( ! [X3: real] :
( ( ord_less_eq_real @ A2 @ X3 )
=> ( ( ord_less_eq_real @ X3 @ B )
=> ? [D4: real] :
( ( ord_less_real @ zero_zero_real @ D4 )
& ! [A4: real,B3: real] :
( ( ( ord_less_eq_real @ A4 @ X3 )
& ( ord_less_eq_real @ X3 @ B3 )
& ( ord_less_real @ ( minus_minus_real @ B3 @ A4 ) @ D4 ) )
=> ( P @ A4 @ B3 ) ) ) ) )
=> ( P @ A2 @ B ) ) ) ) ).
% Bolzano
thf(fact_1260_div,axiom,
tagged6100619406677346166f_real @ ( regular_division @ zero_zero_real @ a @ n ) @ ( set_or1222579329274155063t_real @ zero_zero_real @ a ) ).
% div
thf(fact_1261_intgb__g,axiom,
hensto5963834015518849588l_real @ g @ ( set_or1222579329274155063t_real @ zero_zero_real @ b ) ).
% intgb_g
thf(fact_1262_intgb__f,axiom,
hensto5963834015518849588l_real @ f @ ( set_or1222579329274155063t_real @ zero_zero_real @ a ) ).
% intgb_f
thf(fact_1263_sm__0a,axiom,
monoto4017252874604999745l_real @ ( set_or1222579329274155063t_real @ zero_zero_real @ a ) @ ord_less_real @ ord_less_real @ f ).
% sm_0a
thf(fact_1264_ident__integrable__on,axiom,
! [A2: real,B: real] :
( hensto5963834015518849588l_real
@ ^ [X2: real] : X2
@ ( set_or1222579329274155063t_real @ A2 @ B ) ) ).
% ident_integrable_on
thf(fact_1265_regular__division__division__of,axiom,
! [A2: real,B: real,N: nat] :
( ( ord_less_real @ A2 @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( tagged6100619406677346166f_real @ ( regular_division @ A2 @ B @ N ) @ ( set_or1222579329274155063t_real @ A2 @ B ) ) ) ) ).
% regular_division_division_of
thf(fact_1266_strict__mono__continuous__invD,axiom,
! [A2: real,F: real > real,G: real > real] :
( ( monoto4017252874604999745l_real @ ( set_ord_atLeast_real @ A2 ) @ ord_less_real @ ord_less_real @ F )
=> ( ( topolo5044208981011980120l_real @ ( set_ord_atLeast_real @ A2 ) @ F )
=> ( ( ( image_real_real @ F @ ( set_ord_atLeast_real @ A2 ) )
= ( set_ord_atLeast_real @ ( F @ A2 ) ) )
=> ( ! [X3: real] :
( ( ord_less_eq_real @ A2 @ X3 )
=> ( ( G @ ( F @ X3 ) )
= X3 ) )
=> ( topolo5044208981011980120l_real @ ( set_ord_atLeast_real @ ( F @ A2 ) ) @ G ) ) ) ) ) ).
% strict_mono_continuous_invD
thf(fact_1267_Equivalence__Measurable__On__Borel_Ointegrable__on__mono__on,axiom,
! [A2: real,B: real,F: real > real] :
( ( monoto4017252874604999745l_real @ ( set_or1222579329274155063t_real @ A2 @ B ) @ ord_less_eq_real @ ord_less_eq_real @ F )
=> ( hensto5963834015518849588l_real @ F @ ( set_or1222579329274155063t_real @ A2 @ B ) ) ) ).
% Equivalence_Measurable_On_Borel.integrable_on_mono_on
thf(fact_1268__092_060open_062strict__mono_Aa__seg_092_060close_062,axiom,
monoto4017252874604999745l_real @ top_top_set_real @ ord_less_real @ ord_less_real @ a_seg ).
% \<open>strict_mono a_seg\<close>
thf(fact_1269__092_060open_062uniformly__continuous__on_A_1230_O_Oa_125_Af_092_060close_062,axiom,
topolo8845477368217174713l_real @ ( set_or1222579329274155063t_real @ zero_zero_real @ a ) @ f ).
% \<open>uniformly_continuous_on {0..a} f\<close>
% Helper facts (5)
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $true @ X @ Y )
= X ) ).
thf(help_If_3_1_If_001t__Real__Oreal_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__Real__Oreal_T,axiom,
! [X: real,Y: real] :
( ( if_real @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Real__Oreal_T,axiom,
! [X: real,Y: real] :
( ( if_real @ $true @ X @ Y )
= X ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( ord_less_eq_real
@ ( groups8702937949983641418l_real
@ ^ [K3: set_real] : ( times_times_real @ ( minus_minus_real @ ( f @ ( comple1385675409528146559p_real @ K3 ) ) @ ( f @ ( comple4887499456419720421f_real @ K3 ) ) ) @ ( divide_divide_real @ a @ ( semiri5074537144036343181t_real @ n ) ) )
@ ( regular_division @ zero_zero_real @ a @ n ) )
@ ( groups8702937949983641418l_real
@ ^ [K3: set_real] : ( times_times_real @ ( abs_abs_real @ ( minus_minus_real @ ( f @ ( comple1385675409528146559p_real @ K3 ) ) @ ( f @ ( comple4887499456419720421f_real @ K3 ) ) ) ) @ ( divide_divide_real @ a @ ( semiri5074537144036343181t_real @ n ) ) )
@ ( regular_division @ zero_zero_real @ a @ n ) ) ) ).
%------------------------------------------------------------------------------